Much of the immigration discussion on both nights focused not on how to comprehensively address the issue but a much more narrow concern: whether the government should subsidize health insurance for people who are in the country illegally. Just 38 percent of Americans said that government health care should be available to undocumented immigrants, according to a CNN poll from June; 59 percent said it should not, including almost one-third of Democrats.
As for the proposal to replace private insurance plans with a government-run system, the same poll found the public overwhelmingly disapproved. Fifty-seven percent of those who think the government should provide a national health plan said it should not completely replace private insurance, while 37 percent said it should.
Many Democrats have also expressed concern with the party’s leftward tack on border security as more presidential candidates have expressed support for abolishing the law that makes it a criminal offense to cross the border illegally — an idea that has only recently become a point of disagreement. Many of the more progressive candidates like Ms. Warren and Julián Castro, the former secretary of Housing and Urban Development, have supported decriminalization.
A Marist poll last month of Americans nationwide found that just 27 percent said decriminalization was a good idea, while 66 percent said it was a bad idea. Among Democrats, slightly more disapproved than approved, 47 percent to 45 percent, the poll said.
Prominent Democrats who are raising alarms about the party’s move to the left include veterans of the Obama administration like Jeh Johnson, the former Homeland Security secretary, and Rahm Emanuel.
Mr. Emanuel, the former White House chief of staff, wrote an open letter to the candidates this week imploring them to think beyond the party’s base. “Too often, you succumbed to chasing plaudits on Twitter, which closed the door on swing voters in Wisconsin, Michigan, Pennsylvania, and Ohio,” he wrote.
Republicans said the debates showed Democrats have a lot to worry about.
“They’re basically auditioning to see who’s going to be the sacrificial lamb to Trump,” Mr. Ryun said. Then he added a caveat: “If they keep going down this path, and the economy doesn’t tank.”