CLEVELAND — Speaking to Iowa Republicans here on Monday, Gov. Scott Walker of Wisconsin repeatedly reminded them of his childhood in the state that kicks off the presidential nominating process. Before the same crowd on Tuesday, Senator Tom Cotton of Arkansas had his own boast. “I’m the only politician here this week that married a girl born in Iowa,” said Mr. Cotton, admitting it was a “pander.”

Bruised by their indecorous primary, which left some of the party’s brightest stars reduced to wounded pets on the side of the road and pessimistic about taking over the White House in November, many Republicans are quietly turning their focus to 2020.

Jockeying for future presidential campaigns is a feature of every party convention, and the Republican meeting in Cleveland this week is no exception. But the subtle wooing and chit collecting have taken on a new air of urgency this year, with many party leaders openly skeptical about whether Donald J. Trump can defeat Hillary Clinton.

Mr. Trump’s ascendance has shown that Republican voters craved alternatives to conventional candidates from governor’s mansions and Congress.