Mr. Biden decided against a run four years ago as he mourned his son Beau, who died in May 2015. Now, he is running the kind of campaign that voters might expect from a vice president seeking a promotion immediately after his time as second in command, as if the last four years could be effectively elided. He is promising continuity with his party’s prior administration, pledging to build on successes without wide-scale overhaul and even presenting himself as more of a transitional figure than a long-term standard-bearer.

“I view myself as a bridge, not as anything else,” Mr. Biden said in Detroit on Monday night, campaigning with new endorsers like Gov. Gretchen Whitmer of Michigan and Senators Kamala Harris and Cory Booker. “There’s an entire generation of leaders you saw stand behind me. They are the future of this country.”

Throughout the campaign, he has declined to meet the demands of the loudest voices on the left — he has super PAC support, attends high-dollar fund-raisers and remains unapologetic about overtures to Republicans — and none of it hurt him much. Instead, Mr. Biden is enticing Democrats with a bloodless bargain: a return to normal — or whatever this nation was on Nov. 7, 2016 — and a vow to make Mr. Trump go away, even if Mr. Biden rarely emphasizes a bold new policy vision.

It is a pitch that shares some elements with Mrs. Clinton’s in 2016, centered often on Mr. Trump’s character and behavior. Mr. Biden and his team seem convinced that the reality of Mr. Trump’s tenure will yield a different outcome this time, with the daily tumult more visceral to voters now than it was in Mrs. Clinton’s prescient-but-hypothetical predictions of executive disorder and excessive tweeting.

There is also the messenger to consider. Mr. Biden recently remarked on an “unfair” sexism in the campaign four years ago. Then came an aside, blunt but true: “That’s not going to happen with me.”

Mr. Sanders is summoning 2016 memories of another sort. After his setbacks this month, he finds himself in much the same bind he faced against Mrs. Clinton, seeking ways to prolong the primary but finding few opportunities for true breakthroughs.

In recent days, Mr. Sanders and his allies have made the case that Democrats would be foolish to nominate the more moderate candidate once again, reminding audiences of Mr. Biden’s potential vulnerabilities on matters of trade and the Iraq war.