Yet the backlash to President Trump’s divisive politics has also fueled a demand by the party’s progressive wing for ideological purity and more diverse representation, a tension that could reshape what it means to be a Democrat.

“This is part of the reason Donald Trump won,” Mr. Lipinski said in an interview, adding, “Democrats have chased people out of the party.”

In California, party activists at the state Democratic Convention last week rejected Senator Dianne Feinstein, a moderate lawmaker, refusing to formally bless her re-election. In Texas, the Democratic Congressional Campaign Committee opened fire on a progressive candidate, Laura Moser, posting negative research to blunt her rise in fear that a victory by her in Tuesday’s primary race could doom the party’s bid for a suburban Houston district in November.

But the battle to define the party is playing out most vividly in overwhelmingly safe House districts around cities like Boston, Chicago and New York, where younger liberals, often women, people of color or both, are confronting men who are products of a clubhouse politics where fealty to the organization was paramount.