What to watch for in Tuesday’s primaries.

Voters in Wisconsin, Minnesota, Vermont and Connecticut head to the polls on Tuesday for primaries and for a special election in Minnesota that could further narrow the already tight Democratic majority in the House.

Here is what to watch as the returns roll in:

Another Trump-tinged brawl in Wisconsin.

The expected Democratic primary fight for the right to challenge Senator Ron Johnson, a Republican, fizzled when one contender after another dropped out and endorsed the state’s lieutenant governor, Mandela Barnes, setting up a highly anticipated race for the fall.

But Republicans are in fighting form over who will try to stop the re-election of Wisconsin’s Democratic governor, Tony Evers. Of four remaining candidates, Tim Michel, who has the endorsement of former President Donald J. Trump, and Rebecca Kleefisch, the choice of former Vice President Mike Pence, have the best shot.

If you’re looking for ideological differences, it will be a strain. Ms. Kleefisch, who was lieutenant governor under Gov. Scott Walker, has said President Biden’s victory over Mr. Trump in Wisconsin was “rigged,” though repeated examinations of the nearly 21,000-vote margin have turned up nothing of the kind.

Mr. Michels, an insurance executive for whom Mr. Trump held a rally on Friday, hasn’t ruled out supporting an effort in the Republican-dominated Legislature to overturn the 2020 election results altogether, though such a move would be symbolic at best. He has also promised to abolish the state’s elections commission, a bipartisan regulatory agency that administers elections — a move that Mr. Evers says is meant to give the gerrymandered Legislature the final say in election results.

Both candidates have also taken a hard line against abortion, which, after the repeal of Roe v. Wade, became illegal in the state under a law enacted in 1849. State law now allows abortions only to save the life of the mother, and, during a televised town hall-style debate last week, each of the candidates said they would oppose any additional exceptions.

Given the centrality of Wisconsin in recent presidential elections, and how Mr. Evers has portrayed himself as a Democratic bulwark against Republican efforts to alter the electoral system, much may rest on the outcome of the governor’s race.

A Democratic fight for an open House seat.

In Wisconsin’s only competitive House race, four Democrats are vying to take on Derrick Van Orden, a Republican, in the G.O.P.-leaning seat of Representative Ron Kind, a Democrat who is retiring. Mr. Kind has been a Republican target for years, and Mr. Van Orden came close to beating him in 2020.

Then came Jan. 6, 2021, when Mr. Van Orden, an actor and retired member of the Navy SEALs, was in Washington to protest Mr. Biden’s victory. What exactly he was doing there is in dispute. Democrats say a Facebook photo proved he crossed into a restricted area as Trump supporters stormed the Capitol. Mr. Van Orden denied that and said he was merely in the nation’s capital “to stand for the integrity of our electoral system as a citizen.”

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.

In Vermont, a low-key proxy war for a vacant House seat.

It’s only a single seat, but it is little Vermont’s only one, and the race for the at-large district in the Green Mountain State has pitted two heavyweights against one another in something of a proxy fight for the shape of the left.

Becca Balint, the Vermont Senate president pro tempore, has the vocal support of Senator Bernie Sanders, Vermont’s independent socialist standard-bearer. Her opponent, Molly Gray, is the state’s lieutenant governor, and enjoys the backing of Senator Patrick Leahy, a Democrat.

Mr. Leahy started the shuffling when he announced his retirement this year. The state’s longtime House member, Peter Welch, also a Democrat, quickly lined up the support to move up to the Senate, leaving his House seat to the fighting Democrats.

Notably, even as he endorsed Ms. Gray, Mr. Leahy has repeatedly said that he trusts Vermonters “to make their own decisions about who will represent them.”

But the candidates themselves have seemed to take sides with their endorsers: Ms. Gray said she would serve in the collegial fashion of Mr. Welch and Mr. Leahy, both of whom are Democratic team players. And Ms. Balint, who has the support of a host of big-name progressives, is cut more from the pugilistic cloth of Mr. Sanders, who has promoted her readiness to buck powerful interests, suggesting she may be a less reliable vote for Democratic leadership.

Connecticut’s drama will wait until November.

The Nutmeg State is unusually competitive, thanks to the headwinds facing the Democratic Party. But Tuesday’s primary is largely pro forma, since the tickets for the closest contests are already set.

In the race for governor, Bob Stefanowski, a prominent Republican business executive, is eager for a rematch with Gov. Ned Lamont, a Democrat, who narrowly beat him for the governorship in 2018, a far better year for Democrats.

In the tightest House race this fall, Representative Jahana Hayes, whose 2018 victory made her the first Black woman to represent the state, faces a Republican state senator, George Logan, who is also Black and has the strong backing of Republicans in Washington who are eager to diversify their ranks.

The one real contest that remains in Tuesday’s primary features three Republicans — the Cuba-born Leora Levy, former State Representative Themis Klarides and the Albania-born lawyer Peter Lumaj — vying to challenge Senator Richard Blumenthal. As of now, Mr. Blumenthal does not appear to be in trouble.