Mr. Buttigieg’s critics say he is offering voters feel-good platitudes without a proven track record of electoral success. But his supporters say his vision, and his identity as an openly gay candidate, make him an inherently transformative figure.

He has successfully built a sustainable ground game throughout the state, generating buzz among voters and turning out crowds of several hundred people in towns of just a few thousand. For growing numbers of likely caucusgoers, he is emerging as the moderate front-runner in the race, ahead of even former Vice President Joseph R. Biden Jr. And he is clearly worrying some of his top rivals in the state, like Senator Elizabeth Warren of Massachusetts, who has recently started taking implicit shots at Mr. Buttigieg by needling candidates who have teams of consultants and centrist ideas.

National polls still have Mr. Buttigieg firmly behind Mr. Biden, Ms. Warren and Senator Bernie Sanders of Vermont, and questions remain over Mr. Buttigieg’s ability to cobble together a coalition diverse enough to win the Democratic primary. But the goal of the bus tour, timed to take place after the same Iowa dinner that famously helped create Barack Obama’s presidential mojo in 2007, was to project Mr. Buttigieg as the closest analogue to Mr. Obama currently in the race.

“The idea of hope may have gone out of style a little bit, but when I look at you, I see it is not out of style,” Mr. Buttigieg said in Des Moines. “Are you ready to leave no question about who has the momentum of Iowa right now?”

He has shot up in the polls in Iowa recently and had a one-point edge over Mr. Biden in a recent New York Times/Siena College poll of Iowa voters, within the survey’s margin of error of plus or minus 4.7 percentage points. Ms. Warren held a slight lead in the poll, while Mr. Sanders came in second, just ahead of Mr. Buttigieg and Mr. Biden.