Herman Cain experienced a similar pattern of discovery, scrutiny and decline in October 2011. Mr. Cain’s share of the polls averaged 21 percent before an Oct. 11 debate, but after he emphasized his “9-9-9” tax plan in the debate, his standing in the polls rose to 31 percent. By December, a variety of scandals had helped send him back down to the single digits. He was out of the race not long after.

These moments played a role in the two candidates’ discovery and decline. The debate, however, was not the main driver of opinion change. The way the news media frames debate moments can lead to changes in the polls for candidates. Increasingly, social media amplifies this effect and speeds the process. Once the polls move, a cycle starts that results in more news coverage of a candidate’s gains at the polls, which can then result in even more gains.

What stops most candidates’ post-debate surge is a combination of reality and the heightened focus of media attention, which moves from narratives about an unexpected success or an increase at the polls to scrutiny of the candidate. The focus on a surging candidate may simply bring to light the underlying weaknesses of a candidacy.

Usually, only candidates who are already well known can avoid the decline that scrutiny brings.

Before the dust-up with Ms. Kelly in 2015, Mr. Trump had already been getting an outsize share of mentions in cable news coverage of the election (44 percent of mentions, in a field of 16 other candidates). But after it, his share increased to 64 percent (based on analysis of data from GDELT, which monitors world media, for a book I co-authored about the 2016 election). His standing in the polls improved along with all the increased attention. Mr. Trump had generated a post-debate media boomlet that he leveraged, repeated and rode all the way to the White House.

What explains the difference in public reaction to the scrutiny of Mr. Cain and Mr. Perry compared with that of Mr. Trump? Sometimes information revealed after focal debate moments isn’t really new. Mr. Trump’s penchant for inflammatory language and insults was already known, even if the episode involving Ms. Kelly reached a wider audience and might have seemed at the time to have crossed some line.

The Democratic debates may play out differently for candidates like Joe Biden or Bernie Sanders, whom voters are unlikely to learn much about relative to, say, Kamala Harris and Pete Buttigieg. If the latter two emerge from a good performance with an initial surge in media attention and poll numbers, history suggests that a level of scrutiny will follow that matters more than a good one-liner.

Lynn Vavreck, the Marvin Hoffenberg Professor of American Politics and Public Policy at U.C.L.A., is a co-author of “Identity Crisis: The 2016 Presidential Campaign and the Battle for the Meaning of America.” Follow her on Twitter at @vavreck.