WILMINGTON, Del. — Senator Tom Carper is 13 for 13 in statewide races here, a record of success that dates to his election as state treasurer in 1976. And Mr. Carper’s formula for victory has largely remained constant: stay true to a centrist brand of politics that appeals to Delaware’s mix of rural voters and suburbanites while retaining a ubiquitous presence at churches, firehouse fund-raisers and birthday parties to which he may or may not have been invited.

After sweating his way through a steamy Labor Day parade in downtown Wilmington on Monday, Mr. Carper took even longer to walk back to his Town & Country minivan because he was so eager to discuss his efforts to clean up the city’s waterfront when he was governor of this lightly populated “state of neighbors.” And because he stopped to greet every passer-by, even startling one young couple with an offer to carry their groceries into their apartment.

But as Mr. Carper seeks the Democratic nomination on Thursday for what would be his fourth Senate term, his well-honed strategy is being tested by Kerri Evelyn Harris, a political newcomer whose candidacy reflects the same urgent demands of other progressives in the Trump era. As with many of the generational, racial and ideological challenges rippling through her party, Ms. Harris’s bid is premised on the idea that a platform of pork-barrel projects, congeniality and constituent services is no longer sufficient to the moment.

“A warm smile, a firm handshake and a witty comment isn’t doing it,” said Ms. Harris, who is biracial. “Nice isn’t getting the job done.”