WATERLOO, Iowa—Democrats have an expression for the times when this state has flirted with backing liberals before settling on moderate presidential candidates: “Date Dean, marry Kerry.”

The adage refers to the 2004 caucuses, when then-Sen. John Kerry of Massachusetts defeated that year’s progressive sensation, former Vermont Gov. Howard Dean.

Now, some supporters of moderates Joe Biden and Pete Buttigieg think the same transition is happening, as the liberal candidate who rose in recent months, Massachusetts Sen. Elizabeth Warren, has lost steam in polls in the state.

The movement has resulted in a more aggressive stance by Ms. Warren, especially toward Mr. Buttigieg, the South Bend, Ind., mayor who has recently moved to the top of polls in the first two states to hold nominating contests, Iowa and New Hampshire.

The dynamic is also pitting Mr. Biden, the former vice president, against Mr. Buttigieg, as Minnesota Sen. Amy Klobuchar vies for the center, too. Vermont Sen. Bernie Sanders also ranks in the top tier in polls here.