For Ms. Flores, the question of evolution is key.

“When someone behaves badly, it’s incumbent on them — not us — it’s incumbent on them to change their behavior,” Ms. Flores said in an interview earlier this month. “It doesn’t mean that makes them, necessarily, bad people. The only thing that makes them bad is if they continue the behavior. Joe Biden has demonstrated he’s not very interested in changing his behavior or attitude.”

Mr. Biden’s supporters are keenly aware of that criticism. And they are often eager to loudly dismiss it.

These voters volunteer — sometimes earnestly, sometimes combatively — that they were the ones to request the hugs. They glibly wave off questions about his tactile campaigning style, chalking it up to generational differences, or more often, calling it an asset.

“If you like somebody, you hug them,” said Elizabeth Walek-Sowin, 74, who did just that at a campaign event with Mr. Biden in Henderson, Nevada. “It doesn’t take a Ph.D. to figure that out.”

She proceeded to hug the reporter who had asked about the exchange.

“We’re in a period of time right now where we need hugs,” said Mary McWhorter, 72, after sharing an embrace with Mr. Biden at a campaign stop in Charleston, S.C. “I was thrilled to get to hug Joe Biden. Hell, I’d be happy — he can put his hands on my shoulder any day of the week. I might return the favor, with my husband right here.”

A conversation about hugs often leads to a broader defense of Mr. Biden’s record.

“He has not a racist bone in his body,” insisted Bernadette Brown, a 51-year-old administrative assistant from North Charleston, S.C., soon after explaining why she had hugged Mr. Biden (“that’s just my nature, we greet people, hug people”) at a campaign event after the first debate, where Mr. Biden’s opposition to many busing practices had been a focus. “Racism exists but Vice President Biden did not cause the racism. He is not the problem .”

Mr. Biden is back in South Carolina this week for more campaigning — and perhaps, more hugs, in a state his allies consider to be his firewall.