“They’re doing high-fives in the Clinton campaign right now when they hear this,” Mr. Bush said.

But policy details and disagreements, for the most part, replaced nasty potshots in the early going on Tuesday night, laying bare real fissures within the Republican Party on immigration, national security, trade and the meaning of being a conservative. The candidates used the 90 seconds they were allotted for each answer to promote their tax proposals, to lament what they said were intrusive business regulations and to delve into the country’s monetary policy. Even when Mr. Kasich sought to speak at one point when he was not called on, it was because he wanted to discuss a value-added tax.

Mr. Rubio was not only able to avoid being drawn into the contentious immigration debate, but also repeatedly received questions that allowed him to answer with versions of his stump speech. Even he seemed unable to believe his good fortune when he was asked to make his case against Mrs. Clinton. He chuckled for a moment before unspooling a well-rehearsed argument: why he can prosecute a “generational” case against her.

“If I am the nominee, they will be the party of the past, and we will be the party of the 21st century,” said Mr. Rubio, 44.

Less than three months before Iowa begins the Republican nominating contest, the jostling in Milwaukee reflected the growing urgency the candidates feel to stand out in a sprawling 14-person field. With a pair of political novices, Mr. Trump and Mr. Carson, atop the polls, the other candidates appeared determined to remain in the campaign until there is more clarity on who is likely to be the nominee. But the muddled picture is clearly starting to unnerve many in the party.

Some of the candidates expressed this frustration and sought to distinguish themselves by speaking more aggressively about their plans to cut taxes or create jobs — even if they did not deliver many specifics.

They barely explored recent controversies roiling the Republican race, particularly accusations that Mr. Carson had embellished or even fabricated part of his inspiring life story. Asked if he was worried about those questions, Mr. Carson drew strong applause from the audience by criticizing the news media as biased against him.