Donald Trump showed his enduring grip over Republican primaries for Senate, and establishment Democrats won a House rematch against a progressive challenger.It was an early night in Ohio.Despite questions about turnout amid bad weather, the results of the state’s primary elections on Tuesday didn’t produce many surprises.In the night’s biggest race, J.D. Vance, the “Hillbilly Elegy” author who remade himself as a die-hard supporter of Donald J. Trump, won the closely watched Republican Senate primary after his struggling campaign was lifted by a crucial endorsement from the former president last month.Here are a few key takeaways from one of the first major primary nights of the 2022 midterm cycle:It was a good night for Donald Trump, and not just because of Vance.Mr. Vance’s victory over a crowded field, in which he consolidated support the day of the vote, was unequivocally good news for Mr. Trump. The former president’s endorsement on April 15 came when Mr. Vance had been all but left for dead. Instead, with help from Mr. Trump and allies including Donald Trump Jr., Mr. Vance turned his campaign around.“If Trump supports Vance, then we know he will be good,” said Kurt Oster, 59, a voter in Eaton, Ohio.Trailing Mr. Vance by a relatively wide margin were Josh Mandel, a former Ohio treasurer who had run as a hard-right Trump loyalist — and, like Mr. Vance, faced criticism for contorting himself in doing so — and Matt Dolan, a state senator who sought more moderate voters. Mr. Dolan had seemed to gain ground during early voting, and other campaigns had closely monitored his apparent rise.Josh Mandel giving a concession speech on Tuesday night in Cleveland.Brian Kaiser for The New York TimesBut the fact that Mr. Vance and Mr. Mandel received more than 50 percent of the vote combined running as pro-Trump candidates spoke to the former president’s enduring grip over certain races — particularly Senate primary elections, in which voters are sending people to fight for them in Washington as opposed to run their states.In the general election, Mr. Vance, who improved as a campaigner over the course of the primary, will face Representative Tim Ryan, a moderate Democrat who also claims to understand the concerns of Ohio’s white working class. Part of Mr. Trump’s rationale in endorsing Mr. Vance was his belief that Mr. Ryan would be a strong candidate, and that Mr. Vance was best positioned to take him on, according to a Republican briefed on the endorsement.Mr. Vance speaking to supporters Tuesday night in Cincinnati after his victory.Maddie McGarvey for The New York TimesIt’s not clear how much Mr. Vance’s message will change for the general election in a state that has become increasingly hostile for Democrats. Mr. Ryan, who is trying to win back blue-collar workers for his party, has signaled that he will try to paint Mr. Vance, a Yale Law School graduate and venture capitalist, as a creature of the cocktail party circuit and Silicon Valley. But he faces an uphill battle in Ohio.For governor, Republican voters in Ohio preferred a familiar face.The night did not completely belong to Mr. Trump and Trumpism.Gov. Mike DeWine easily won the Republican nomination for another term despite angering many in the Trump wing of the party for what they saw as his heavy hand in controlling the pandemic. Last month, Mr. DeWine said that he could not attend a Trump rally in his state because he was committed to celebrating Ulysses S. Grant’s 200th birthday.Gov. Mike DeWine and his wife, Fran, after voting on Tuesday in Cedarville, Ohio.Paul Vernon/Associated PressHis main opponent, Jim Renacci, sought out Mr. Trump’s endorsement but did not secure it, in large part because he was never a serious threat. Mr. Renacci’s “Ohio First” campaign was clearly an echo of Mr. Trump’s presidential bids, yet he never gained traction.A Trump ally rose, as a Republican who backed impeachment departs.One of Mr. Trump’s other victories in Ohio was that of Max Miller, a young former aide who worked for him in the White House.With Mr. Trump’s encouragement, Mr. Miller ran for Congress in a state where his family has deep ties, initially as an attempt to take out a House Republican who had voted to impeach Mr. Trump after the Capitol riot. That congressman, Anthony Gonzalez, dropped out. But when the seats were redrawn during redistricting, Mr. Miller ran in a different district, and won his primary on Tuesday night.Despite some ugly headlines — Mr. Miller was accused of domestic violence by an ex-girlfriend, Stephanie Grisham, one of Mr. Trump’s press secretaries, an allegation that he denied before suing for defamation — he is expected to carry the safely conservative district easily in November.And if he does win, another House member whose candidacy began as a vengeance play will owe his political rise to the former president.It’s better to be the only Trump acolyte than the only establishment Republican in a race.Splitting the pro-Trump vote didn’t save Mr. Dolan’s candidacy in the Senate primary, but splitting the establishment Republican vote handed a pro-Trump candidate a surprising victory in Northwest Ohio’s Ninth Congressional District.J.R. Majewski, a burly businessman who painted his vast back lawn into one huge Trump sign in 2020, earned the right to challenge Representative Marcy Kaptur, a Democrat who has served in Congress for decades. Her district was redrawn by the state’s Republican-controlled Legislature to try to thwart her bid for a 21st term.The new boundaries attracted two G.O.P. state lawmakers, State Senator Theresa Gavarone and State Representative Craig Riedel, to enter the primary. Then, almost as an afterthought, came Mr. Majewski, who ran ads showing him carrying an assault-style rifle, posted a “Let’s Go Brandon” rap on his website and earned a somewhat incoherent acknowledgment from Mr. Trump at an Ohio rally.The battle between Ms. Gavarone and Mr. Riedel, however, appeared to let Mr. Majewski squeeze through — though Ms. Kaptur may get the last laugh.Ohio Democrats showed little appetite for adding a new ‘squad’ member.Last August, Shontel Brown, a little-known chairwoman of the Cuyahoga County Democratic Party, seemingly came from nowhere to win a House special election in Cleveland against Nina Turner, a former co-chair of Senator Bernie Sanders’s presidential campaign and a hero of the activist left.In Washington, the Democratic establishment had dearly wanted to keep Ms. Turner away from the House. She had made something of a career of bashing centrist Democrats, and planned to be a brash voice in the expanding “squad” of progressive members of Congress. Ms. Brown was seen by many on the left as the establishment’s creation.Representative Shontel Brown at her watch party in Cleveland on Tuesday night.John Kuntz/Cleveland.com, via Associated PressMs. Turner surprised no one when she challenged Ms. Brown to a rematch in this year’s Democratic primary.Her pitch was that this year would be different. Crossover Republicans from the Cleveland suburbs who had helped Ms. Brown in the special election would not be available this time, because they would be voting in the Republican primary. A redrawn district, still overwhelmingly Democratic, was more concentrated in and around Cleveland, Ms. Turner’s home base.But Ms. Brown ran this year not as an unknown but as an incumbent, who could point to her vote for the bipartisan infrastructure law. The Congressional Progressive Caucus endorsed her, blunting any boost Ms. Turner might have received from Mr. Sanders’s endorsement and late support from Representative Alexandria Ocasio-Cortez of New York.And in a disheartening blow for Ms. Turner and the activist left, Ms. Brown easily won the rematch.Kevin Williams More