MILWAUKEE — Democrats formally nominated Senator Kamala Harris for the vice presidency on Wednesday night, placing a woman of color on a major party ticket for the first time and showcasing the diversity of race and gender they believe will energize their coalition to defeat President Trump in the fall. The third night of the party’s national convention also featured a striking repudiation of Mr. Trump by former President Barack Obama, a break with the presidential custom of not criticizing a successor by name. Mr. Obama praised Mr. Biden’s character, contrasting it with Mr. Trump’s, and directed a portion of his remarks to voters undecided about whom they will vote for, or whether they will vote at all. “Donald Trump hasn’t grown into the job, because he can’t,” Mr. Obama said, growing emotional at points as he talked about the challenges facing the country and democracy. “The consequences of that failure are severe. One hundred seventy thousand Americans dead. Millions of jobs gone.” A day after nominating Joseph R. Biden Jr., a 77-year-old fixture of Washington establishment politics who served as Mr. Obama’s vice president, Democrats tried to make the case that while Mr. Biden would be one kind of change agent — a repudiation of Trumpism — Ms. Harris would help steer the party in new directions and reflect a changing America. Speeches by Mr. Obama, former Secretary of State Hillary Clinton, Senator Elizabeth Warren and Speaker Nancy Pelosi were intended to underscore the history-making moment of Ms. Harris’s nomination, highlighting her uniquely American biography: A child of immigrants and a graduate of a historically Black university, she is one of the few women of color elected to the United States Senate. “We’re at an inflection point,” Ms. Harris, 55, said as she formally accepted the nomination. “The constant chaos leaves us adrift. The incompetence makes us feel afraid. The callousness makes us feel alone. “We can do better and deserve so much more. We must elect a president who will bring something different, something better and do the important work.” Far more than the two previous nights, which centered on testimonials to Mr. Biden’s character and empathy, the program focused on policy, addressing issues like gun violence, climate change, affordable child care and immigration. In videos, activists promoted Mr. Biden’s plans to tackle a warming planet, and survivors of sexual assault and domestic violence reminded viewers of his role in crafting the Violence Against Women Act. The American child of a deported undocumented mother begged the president to reunite families torn apart by his immigration policy. In perhaps the most policy-heavy speech of the evening, Ms. Warren, speaking from an early childhood learning center in her home state of Massachusetts, praised Mr. Biden’s “really good plans.” She highlighted his proposals to make child care more affordable, to provide universal preschool and to raise wages for child care workers. Much of the evening was devoted to the power of women in politics. In a week marking the hundredth anniversary of the 19th Amendment, which gave women the right to vote, Ms. Pelosi and Mrs. Clinton, two of the most influential women in politics, wore white as they delivered their remarks, in homage to the suffragists. Both anointed Ms. Harris as a successor of sorts, though they had declined to endorse her or any of the other five women who sought the Democratic presidential nomination during the primaries. ImageApplause was broadcast on a video screen at the site of Ms. Harris’s speech. The convention has been conducted virtually because of the coronavirus pandemic.Credit...Erin Schaff/The New York Times Yet, even as Democrats championed change, they promised to keep fighting for policies to combat sexual assault and domestic violence and to improve access to affordable child care. The prominent airtime given to those issues underscores the influence Democratic women have gained during the Trump era. Women have emerged as the backbone of the party, shattering records for political giving, running for office in unprecedented numbers and overwhelmingly voting for Democrats. Wednesday night’s program, conducted virtually because of the coronavirus pandemic, was a tribute to the constituencies that have driven the party’s rise during the Trump administration — women, minority voters, and young voters. While Ms. Harris was the evening’s main attraction, the program featured remarks from several of the most powerful women in the party, as well as Spanish-language speakers, victims of gun violence and everyday Americans meant to represent marginalized slices of the electorate. Like the previous evenings, the night combined elements of an old-fashioned variety show, a telethon and a political event. The most unexpected development of the night came from Mr. Obama, whose condemnation of Mr. Trump was a departure from his first turn on the convention stage, in 2004, when he catapulted to national prominence with a soaring manifesto of hope and national unity. On Wednesday, his message was darker, a reflection of a country changed by crisis and a party desperate to oust an incumbent president. He offered a grim warning about the durability of American democracy. “This president and those in power — those who benefit from keeping things the way they are — they are counting on your cynicism,” Mr. Obama said, addressing voters. “They know they can’t win you over with their policies. So they’re hoping to make it as hard as possible for you to vote, and to convince you that your vote doesn’t matter. That’s how they win.” Mr. Obama has largely held his fire for three and a half years as Mr. Trump has gone after him relentlessly with fierce attacks and baseless smears. And the former president has mostly stayed out of American politics, other than occasionally offering advice and endorsements to Democratic candidates. Many supporters have long wished that he would speak out against Mr. Trump; on Wednesday he did, focusing chiefly on the Republican’s approach to the pandemic, leadership and democracy. He said the effects of the coronavirus crisis had made clear the stakes of stable leadership, and he cited the American deaths from the virus that Mr. Trump has rarely acknowledged. He also alluded to some of Mr. Trump’s most controversial moments, saying his actions had eroded America’s standing throughout the world. Mr. Obama delivered his remarks from the Museum of the American Revolution in Philadelphia, in a room with the U.S. Constitution displayed on its walls. The subtext was clear: The core of the nation’s democracy was at stake. “Do not let them take away your power,” Mr. Obama said. “Do not let them take away your democracy.” Video transcript Back transcript Presidency Is ‘One More Reality Show’ to Trump, Obama Says Barack Obama criticized President Trump in his comments at the Democratic National Convention, saying Mr. Trump had shown “no interest” in using the office to help anyone but “himself and his friends.” Tonight I want to talk as plainly as I can about the stakes in this election — because what we do these next 76 days will echo through generations to come. I have sat in the Oval Office with both of the men who are running for president. I never expected that my successor would embrace my vision or continue my policies. I did hope for the sake of our country that Donald Trump might show some interest in taking the job seriously, that he might come to feel the weight of the office and discover some reverence for the democracy that had been placed in his care. But he never did. For close to four years now, he has shown no interest in putting in the work, no interest in finding common ground, no interest in using the awesome power of his office to help anyone but himself and his friends, no interest in treating the presidency as anything but one more reality show that he can use to get the attention he craves. Donald Trump hasn’t grown into the job, because he can’t. And the consequences of that failure are severe. This administration has shown it will tear our democracy down, if that’s what it takes for them to win. So we have to get busy building it up — by pouring all our efforts into these 76 days and by voting like never before: for Joe and Kamala and candidates up and down the ticket, so that we leave no doubt about what this country that we love stands for. Barack Obama criticized President Trump in his comments at the Democratic National Convention, saying Mr. Trump had shown “no interest” in using the office to help anyone but “himself and his friends.”CreditCredit...Democratic National Convention In her acceptance speech, Ms. Harris wove her personal story with policies that she said would improve the lives of all Americans “to achieve the future we collectively want.” Ms. Harris, who is of Jamaican and Indian descent, said her experiences would ensure that the perspective of people long marginalized in America — African-Americans, Asian-Americans, women, first-generation residents — would have a voice at the highest levels of Mr. Biden’s administration. She said she was committed to “a vision of our nation as a beloved community — where all are welcome, no matter what we look like, where we come from, or who we love.” While the pandemic denied Ms. Harris, California’s junior senator, the typical pomp that awaits any vice-presidential nominee — a clamoring convention audience of thousands — it provided her a controlled setting to reintroduce herself to the American people after her unsuccessful presidential campaign. She delivered her remarks from Mr. Biden’s home state of Delaware — rather than from Milwaukee, where the convention was initially chosen to take place — and she was introduced by her sister and niece, both of whom played key roles in her primary campaign, and her stepdaughter. Though voters selected Mr. Biden as their nominee over candidates who made up the most diverse slate ever in a presidential primary race, he has often been cast as a transitional figure, selling himself as an elder statesman who can stabilize a country in crisis and restore a sense of decency to government. Ms. Harris, meanwhile, was positioned Wednesday night to represent the diverse electorate that Democrats have long maintained will enshrine them in power, a forward-looking complement to Mr. Biden’s steady hand. Many Democrats have speculated about the possibility of Mr. Biden’s serving only one term if elected, and the selection of Ms. Harris as his running mate installs her as the leading candidate for 2024 should he choose not to run again. Mrs. Clinton made the case for Mr. Biden and Ms. Harris as a team offering policy and empathy, saying they could move the country forward in areas such as housing and health care. She noted that Ms. Harris would face attacks as a woman on the national ticket, but added, “I know something about the slings and arrows she’ll face, and believe me, this former district attorney and attorney general can handle them all.” Making an urgent plea for Americans to turn out for this election, Mrs. Clinton said: “This can’t be another woulda, coulda, shoulda election. No matter what, vote. Vote like our lives and livelihoods are on the line — because they are.” On Wednesday, the program offered a diverse lineup of speakers, a majority of whom were women. Former Representative Gabrielle Giffords of Arizona, one of the country’s most vocal advocates for stricter gun laws, delivered a moving plea for curbing gun violence. Other speakers addressed climate change, immigration and economic policy. An emotional video featuring an undocumented family in North Carolina was filmed entirely in Spanish, with English subtitles. In a halting speech, Ms. Giffords, who has struggled with speaking since being shot in the head at a congressional event in 2011, urged viewers to support Mr. Biden. “Words once came easily, but today I struggle to speak. But I have not lost my voice,” she said in a pretaped video. “We must elect Joe Biden. He was there for me. He’ll be there for you, too.” Video transcript Back transcript ‘This Can’t Be Another Woulda, Coulda, Shoulda Election,’ Clinton Says On the third night of the Democratic National Convention, Hillary Clinton endorsed Joe Biden and made comedic reference to her own presidential campaign against Donald J. Trump. Trump came in with so much set up for him: a strong economy, plans for managing crises including a pandemic. Yes, we Democrats would have disagreed with him on many things. But if he had put his own interests and ego aside, seen the humanity in a child ripped from her parents at the border, or a protester calling for justice, or a family wiped out by natural disaster, that would have been a good thing for America and the world. For four years, people have told me: “I didn’t realize how dangerous he was. I wish I could do it all over.” Or worst, “I should have voted.” Look, this can’t be another woulda, coulda, shoulda election. If you vote by mail, request your ballot now and send it back right away. If you vote in person, do it early. Become a poll worker. Most of all, no matter what, vote. As Michelle Obama and Bernie Sanders warned us, if Trump is re-elected, things will get even worse. Vote for honest elections — so we, not a foreign adversary, choose our president. Vote for the diverse, hopeful America we saw in last night’s roll call. And don’t forget: Joe and Kamala can win by 3 million votes and still lose. Take it from me. On the third night of the Democratic National Convention, Hillary Clinton endorsed Joe Biden and made comedic reference to her own presidential campaign against Donald J. Trump.CreditCredit...Democratic National Convention Yet Ms. Harris was the night’s unquestioned focus, showing signs of the political prowess that once made her a top-tier presidential candidate. During her campaign, she struggled to find a balance between the party’s progressive and moderate wings, and failed to deliver a consistent message on policy. While Democratic activists and political insiders pressured Mr. Biden to showcase his commitment to diversity by selecting a Black or Latino running mate, Democratic voters have rarely placed the same importance on representation, even at a moment of rapidly changing national views on race and gender. During the primary race, many Black voters expressed disenchantment with the idea that racial representation equated to change, and that they should automatically support a candidate who looked like them. But with Ms. Harris on the ballot, some Democrats believe that turnout among minority voters may be higher, and that the same voters who overlooked her as a presidential candidate will be inspired by her as a running mate. But for all the embrace of Ms. Harris for symbolizing a new era of Democratic politics, there was little on Wednesday night to suggest that her ascension would lead to a break from mainstream party orthodoxy. A former prosecutor and California attorney general, she supported ideas like the expansive Green New Deal to combat climate change during her presidential bid but rejected left-wing litmus tests such as “Medicare for all” and higher taxes for the wealthiest Americans. On issues such as racial equality, her proposed solutions mimicked the policies of Mr. Obama and Mrs. Clinton, not those of the activists who pushed the Democratic Party leftward in recent years. That did not stop Mr. Trump from trying to use her nomination as a moment to sow division. “BUT DIDN’T SHE CALL HIM A RACIST??? DIDN’T SHE SAY HE WAS INCOMPETENT???” the president tweeted, one of several all-caps messages he sent during her and Mr. Obama’s remarks. Ms. Harris herself had a few zingers for Mr. Trump, harking back to her years as a prosecutor fighting gangs, big banks and for-profit colleges. “I know a predator when I see one,” she said.