But the fissures within the Democratic Party had senior Trump campaign officials predicting that after the debate they would see a bump in their poll numbers in critical Rust Belt states like Michigan and Pennsylvania. In a statement released Thursday, Brad Parscale, Mr. Trump’s campaign manager, said the debates showed the Democrats’ “vendetta against coal, oil, and natural gas.” The campaign believes that will alienate voters in those key states, which Republicans had not won in decades until Mr. Trump carried them in 2016.

Though they were realistic about their limitations — and the fact that the general election is more than a year away — Republicans said that the debates had given them fodder for what they say will be hundreds of millions of dollars of attack ads.

“Thank you very much for what you are doing, because you are giving us unlimited amounts of video footage for us to use,” said Ned Ryun, the chief executive of American Majority, a group that is training and recruiting conservatives to help turn out the vote for Republicans in battleground states.