In excerpts from paid speeches to financial institutions and corporate audiences, Mrs. Clinton embraced unfettered international trade and offered praise for a budget-balancing plan that would have required cuts to Social Security. She spoke of the need for “a public and a private position” on politically sensitive issues. And she allowed that her family’s growing wealth had left her “kind of far removed” from the experience of the middle class.

“I feel like I’m channeling Captain Renault from ‘Casablanca,’” said Jonathan Tasini, a former union leader who challenged Mrs. Clinton in her Senate primary in New York in 2006. “I’m shocked — shocked! — that Hillary Clinton has a close relationship with Wall Street.”

It is a familiar, if still painful, sensation for Sanders backers, even as most of his voters drift toward Mrs. Clinton, some more haltingly than others.

For at least a handful, the emails have especially rankled given the seeming free fall of Mr. Trump, which has bolstered their view that Mr. Sanders’s proudly left-wing politics would not have precluded victory in the general election.

On the heels of leaked emails over the summer from the Democratic National Committee, which suggested favoritism toward Mrs. Clinton among party leaders, and persistent complaints that Mr. Sanders’s bid was not taken seriously enough from the start, Sanders allies say the latest revelations have heightened tensions that are likely to persist if Mrs. Clinton is elected.

“There is still this real disconnect between her and working people. That’s very difficult to see,” Winnie Wong, a founder of People for Bernie, a group of Sanders supporters, said of Mrs. Clinton. “The people really have to get together and make sure some of these agenda items on the platform become a reality.”

Ms. DeMoro, of the nurses’ union, expressed outrage over an email sent to Mr. Podesta by Randi Weingarten, the president of the American Federation of Teachers, who vowed to combat the nurses’ “high and mighty sanctimonious conduct.” (Mr. Podesta replied, “Thanks.”)