Primary victories for Trump allies in Arizona and Michigan on Tuesday reaffirmed the former president’s continued influence over the Republican Party, as he has sought to cleanse the party of his critics, install loyalists in key swing-state offices, and scare off potential 2024 rivals with a show of brute political force.

Mr. Trump’s choice for Senate, Blake Masters, won a crowded primary in Arizona, as did his choice for Secretary of State, Mark Finchem, an election denier who has publicly admitted his affiliation with the far-right Oath Keepers militia group. Even though Mr. Trump’s choice, Kari Lake, was badly outspent, the governor’s race was virtually tied early Wednesday.

In a particularly symbolic victory for Mr. Trump, Rusty Bowers, the Republican speaker of the Arizona House who gained national attention after testifying against Mr. Trump during the congressional hearings on Jan. 6, lost his bid for State Senate.

Representative Peter Meijer, a House Republican who voted to impeach Mr. Trump, was defeated by a former Trump administration official, John Gibbs, in Michigan, and Mr. Trump’s last-minute choice for governor, conservative commentator Tudor Dixon, who has echoed his false claims of election fraud, easily won her primary.

Mr. Trump and his supporters have been particularly focused on the vote-counting and certification processes in both Arizona and Michigan, seeking to oust those who stood in their way of overturning the 2020 election. Mr. Finchem’s victory, which he celebrated by marching on the Capitol on Jan. 6, was a key indicator of how the “Stop the Steal” movement, which was founded on a falsehood about 2020, has evolved into a widespread campaign to seize control of the levers of democracy ahead of the upcoming elections.

The primaries in five states on Tuesday — Arizona, Michigan, Missouri, Kansas, and Washington State — kicked off a final six-week stretch of races that will provide the most complete picture of the Republican Party’s priorities in 2022, how tight Mr. Trump’s hold on the base remains, and the extent to which his lies about a stolen election in 2020 have infected the electorate.

In Washington State, Mr. Trump had backed challengers to two Republican House members who voted for his impeachment. But both of those incumbents appeared to be in strong positions to advance over Mr. Trump’s preferred candidates — benefiting from the state’s top-two primary system, though neither race had been called early Wednesday.

Missouri Republicans rejected Eric Greitens, the scandal-plagued former governor who ran for Senate, providing relief to national party strategists. Party leaders were concerned that Mr. Greitens would jeopardize Republicans’ otherwise secure Senate seat. Mr. Trump had stayed out of the race until Monday’s bizarre last-minute dual endorsement of “Eric” — with no last name — a blessing that covered both Mr. Greitens, who finished third, and Eric Schmitt, the state attorney general, who won the Senate nomination.

In Kansas, voters sent a warning signal to bullish Republicans, as a ballot measure on abortion demonstrated the issue’s electoral potency and shifting politics in the aftermath of the Supreme Court’s ruling overturning Roe v. Wade.  Voters there overwhelmingly rejected an attempt to amend the state constitution to remove the constitutionally protected right to abortion.

On Tuesday, several of the marquee Republican races were in Arizona, a top presidential battleground state with an open governor’s race, a contested Senate seat, and multiple competitive House races in 2022.

Mr. Trump endorsed Ms. Lake in the governor’s race, a telegenic former newscaster who had become an unabashed supporter of Trumpism. Mr. Trump is looking for redemption after struggling in other governor’s races earlier this year, most notably failing to unseat Republican Governor Brian Kemp of Georgia.

Unlike Mr. Kemp, the Republican governor of Arizona, Doug Ducey, who enraged Mr. Trump by endorsing the results of the 2020 election, was term-limited and did not appear on the ballot. Mr. Ducey backed Karrin Taylor Robson, a wealthy real estate developer who spent more than $18 million on her campaign and who also had the support of Mr. Trump’s former vice president, Mike Pence.