The dynamics pose particular problems for Pete Buttigieg, the mayor of South Bend, Ind., and to some extent Senator Elizabeth Warren of Massachusetts: Though both candidates are in the top tier in Iowa, they are struggling to connect with black voters, the most critical voting bloc for Democrats, fueling the perception that they appeal mainly to liberal white elites.

A recent Quinnipiac University poll showed Mr. Buttigieg with less than 1 percent support among black Democrats in South Carolina; Ms. Warren had 8 percent. Another Quinnipiac poll, released last week, had Mr. Buttigieg with 2 percent support nationally among black voters and Ms. Warren with 12 percent. Senator Bernie Sanders of Vermont had 13 percent.

Those weaknesses are now giving some Iowans pause.

“That matters to people in Iowa because they think about things strategically,” said Douglas Burns, an owner of the Carroll Times Herald in northwestern Iowa. “We view ourselves as representatives or ambassadors for the rest of the country — we represent ourselves but we’re also proxies for other people.”