President Trump’s chosen candidates swept three Republican Senate primaries on Tuesday, but he ran into trouble in other races in Georgia.

Mr. Trump’s pick for governor of Georgia, Lt. Gov. Burt Jones, lost the Republican primary runoff to Rick Jackson, a wealthy health care executive. It was the second time this month that the president had backed the losing candidate in a Republican primary for governor, following the defeat of Representative Randy Feenstra in Iowa.

The president did pick up a key victory in Georgia: Representative Mike Collins, an immigration hard-liner, won his primary to face Senator Jon Ossoff in what is expected to be a bruising and expensive contest. And Mr. Trump’s picks for Senate also prevailed in deeply Republican Alabama and Oklahoma.

Here are six takeaways from Tuesday night.

Georgia Republicans are still willing to buck Trump …

Mr. Jones, whose loss on Tuesday was probably the biggest upset of the evening, worked with Trump allies to try to overturn his loss in the 2020 presidential election. And an election denier lost the runoff for Georgia secretary of state, an indirect rebuke of the president’s baseless claim that the 2020 vote was rigged.

Mr. Trump’s history of being disappointed with Georgia Republicans goes back to 2020, when Gov. Brian Kemp and Brad Raffensperger, the secretary of state, defended Georgia’s presidential election results. And in early 2021, Mr. Trump’s picks for both of Georgia’s Senate seats lost their runoff elections, launching Mr. Ossoff’s political career.

… but the president did better elsewhere.

Mr. Trump fared better with his picks for the Senate. Mr. Collins, a two-term congressman in Georgia and Representative Barry Moore, a three-term congressman in Alabama, both defeated competitors who were running as outsiders. Representative Kevin Hern, the Trump-backed Senate contender in Oklahoma, also advanced to the general election.

Those victories showed Mr. Trump’s enduring strength with Republican primary voters in conservative states, even as the president’s overall popularity with Democrats and independents has fallen. In an interview, Mr. Moore credited Mr. Trump for his victory.

“I can’t tell you how many people I walked up to, especially older people from Alabama, and they said, ‘If the president’s with you, we’re with you,’” he said.

To win as a political outsider, it helps to be fabulously wealthy.

Money does not always win elections (see: Tom Steyer’s costly Democratic bid for California governor this year). While the campaigns of Mr. Jackson and Mr. Jones spent $162 million in the Republican primary for Georgia governor, Mr. Jackson’s total was more than double that of his opponent, according to data from AdImpact, a media tracking firm.

All those ads helped Mr. Jackson introduce himself as a political outsider and deliver his message that he embraced Trumpism, even without the president’s endorsement. He would be like Mr. Trump, he said, “with a Southern tone.”

Mr. Jackson also cut an ad featuring the governor, Mr. Kemp, praising him in a video interview. It suggested that he had the term-limited governor’s support even though Mr. Kemp ultimately endorsed Mr. Jones. Speaking of which …

Brian Kemp had a tough night.

Mr. Kemp has been popular in his two terms as governor, but two of the candidates he backed lost on Tuesday night. Mr. Jones was defeated in the race to succeed him.

And Derek Dooley, a lawyer and former football coach Mr. Kemp had recruited to run for Senate after he decided not to seek the seat himself, faltered as well. Mr. Kemp had argued that Mr. Dooley, as an outsider, would have the better shot at defeating Mr. Ossoff in November.

But his pitch did not resonate with enough Republican voters, who have tended to prefer candidates they see as dependable fighters.

An election denier will not be running Georgia elections.

Tuesday’s results offered one bright spot for Mr. Kemp: One of his former aides, State Representative Tim Fleming, won the Republican primary runoff to become Georgia’s next secretary of state. Mr. Fleming defeated Vernon Jones, a former DeKalb County chief executive and ex-Democrat, who has falsely claimed that the 2020 presidential election was stolen. Mr. Fleming has described that election has having had irregularities.

Mr. Fleming will compete in a November election that could become a referendum on the 2020 vote in Georgia, which both Mr. Kemp and Mr. Raffensperger have declared emphatically that Mr. Trump lost.

Mr. Ossoff has dismissed both Mr. Collins and Mr. Dooley as “puppets” of the president. But he worked quietly for months to undermine Mr. Dooley, who Democrats and Republicans alike thought would appeal to more independent and crossover Democratic voters.

The map of Tuesday’s G.O.P. primary results shows why: Mr. Collins’s biggest strength was with voters outside the Atlanta area, while Mr. Dooley performed more strongly in the more populous suburban counties that helped Democrats win both of the state’s Senate seats in recent years.

After Mr. Collins’s victory on Tuesday, Mr. Ossoff harshly criticized him in a social media post, calling him a “notorious bigot, antisemite and extremist.” He also pointed to a House Ethics Committee inquiry into his office that Mr. Collins has dismissed as “bogus.”

In his victory speech, Mr. Collins also got in digs at Mr. Ossoff, calling him a “far-left liberal” weak on immigration and crime. “This race is going to be challenging,” he added.

Reporting was contributed by Tim Balk, Emily Cochrane, Reid J. Epstein, Shane Goldmacher, Sean Keenan, Rick Rojas and Theodore Schleifer.