What to watch for in Tuesday’s primaries.

All of his would-be Democratic opponents say Mr. Van Orden is singularly dangerous, but they are taking different tacks. Brad Pfaff, a Wisconsin state senator, is emphasizing his accumulated experience in politics and claiming the mantle of Mr. Kind’s style of centrism. His main rival, Deb McGrath, a former C.I.A. officer and Army captain, is taking a more liberal line and a more confrontational stance against Mr. Van Orden, a fellow veteran who she said “took the same oath that I did to protect and defend the Constitution,” but then went to Washington on Jan. 6.

Whoever wins the primary can expect an uphill climb to keep the seat in Democratic hands.

In Minnesota, a special election could narrow the Democratic edge in the House.

The death from cancer of Representative Jim Hagedorn, a Republican, in February has given voters in his southern Minnesota district the chance to fill his seat just in time for the final big votes in an already narrowly divided House. In another year, the seat might have been more competitive. After all, it was represented by Tim Walz, now Minnesota’s Democratic governor, before Mr. Hagedorn was elected in 2018.

But the district leans Republican, and Brad Finstad, a farmer who held an Agriculture Department post in the Trump administration, has the edge over his Democratic opponent, Jeff Ettinger. Mr. Ettinger, the former chief executive of Hormel Foods, is not making it easy on the Republicans: He has poured $900,000 of his own money into the race. Mr. Finstad, who won his May primary by just 427 votes, must unite the G.O.P. and turn out the vote in a state where the big-ticket race for governor does not have a competitive primary.

In an odd quirk, Republican voters in the district will cast two ballots, one to determine who will serve in the House through the end of the year, and another for a primary to choose the party’s candidate in November, when the district selects its representative for the next Congress.

A Finstad victory in the special election would narrow the Democratic margin of control in the House to just four votes. In the primary vote, meanwhile, Mr. Finstad faces the man he barely beat in May, Jeremy Munson. That raises the intriguing possibility that Mr. Finstad could win one race and lose another on the same day.