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Former President Donald Trump鈥檚 crusade for vengeance suffered two devastating blows after Georgia Gov. Brian Kemp and Secretary of State Brad Raffensperger won their primaries Tuesday despite rejecting Mr. Trump鈥檚 entreaties to reverse his 2020 election loss.

It鈥檚 a huge warning sign for the way Republican voters view the former president鈥檚 crusade to punish those who were not willing to overturn the will of the voters in 2020.

Voters also demonstrated an openness to embracing scandal-plagued candidates聽鈥� depending on the candidate, and the scandal.

Here are some takeaways from Tuesday鈥檚 primary elections in Georgia, Alabama, Arkansas, Texas, and Minnesota:

Trump鈥檚 biggest primary defeat

Mr. Trump had hoped to turn Georgia Gov. Brian Kemp into an example of the danger in defying him. Instead, Mr. Kemp on Tuesday became an example of how Republican incumbents might not have as much to fear from Mr. Trump as the former president would like.

Mr. Kemp cruised past former U.S. Sen. David Perdue in the Republican primary. The victory came a year and a half after Mr. Kemp rejected Mr. Trump鈥檚 demands to help overturn the presidential election by declaring Mr. Trump the winner in Georgia instead of Joe Biden, who actually won.

Mr. Perdue鈥檚 campaign fixated on Mr. Trump鈥檚 lie that the 2020 presidential election was stolen from him, but Mr. Kemp won by flexing the power of his office. To rally the base, he signed laws allowing most Georgians to carry guns without a permit and banning most abortions after a fetal heartbeat is detected. He also announced an investment by Hyundai in a new plant in the state to make batteries for electric vehicles.

Now Mr. Kemp will face Democrat Stacey Abrams in a rematch of their 2018 gubernatorial clash. Unlike Mr. Trump in 2020, Mr. Perdue accepted his defeat Tuesday night, even seeming to brush aside some supporters who took up a chant suggesting there was fraud.

鈥淚鈥檓 sorry, but what we鈥檙e going to do right now is make sure Stacey Abrams is not governor of this state,鈥� Mr. Perdue said.

Danger of election denial

The Georgia governor鈥檚 race wasn鈥檛 the only Trump grudge match that backfired on the former president. Secretary of State Brad Raffensperger, who personally rejected Mr. Trump鈥檚 call to 鈥渇ind鈥� enough votes to declare him the winner in Georgia, defeated his Trump-backed primary challenger as well.

Mr. Trump recruited U.S. Rep. Jody Hice from a safe congressional seat to face Mr. Raffensperger in the Republican primary, but Mr. Hice lost. Mr. Trump endorsed primary challengers to the insurance commissioner and attorney general, and they, too, lost.

It鈥檚 clear the former president鈥檚 harping on 2020 simply did not speak to Republican voters in Georgia, the country鈥檚 newest battleground state.

鈥淕eorgia underscores one of Trump鈥檚 big problems if/when he runs again,鈥� Brendan Buck, a former spokesperson for onetime House Speaker Paul Ryan, tweeted Tuesday. 鈥淗e, of course, won鈥檛 be able to let go of the 2020 nonsense, and nobody wants to hear his whining about it anymore.鈥�

Mr. Trump has scored some primary victories with election deniers聽鈥� most significantly last week in Pennsylvania, when Republican voters there chose his preferred gubernatorial candidate Doug Mastriano, who said he wouldn鈥檛 have certified Mr. Biden鈥檚 2020 win of the state.

But multiple Republicans have made clear they鈥檙e eyeing 2024 presidential bids, including Mr. Pence and Florida Gov. Ron DeSantis. And they have distanced themselves in ways large and small from Mr. Trump鈥檚 election allegations. Elections are usually about the future, and by the time the 2024 GOP primary rolls around, November 2020 will be ancient history.

Declining power of scandal

Mr. Trump did notch some wins Tuesday. They came with baggage, but that didn鈥檛 seem to stop them.

Former football great Herschel Walker, Mr. Trump鈥檚 pick for U.S. Senate in Georgia, dominated his Republican challengers. Party leaders had first shied away from him because of his checkered history.

Mr. Walker, in his autobiography, admits struggling with mental illness. His ex-wife said that during their marriage he held a gun to her head and threatened to kill her. He claimed to have founded a chicken processing company employing hundreds but reported only eight workers when applying for a loan during the coronavirus pandemic. He lied about founding a charity to help veterans get aid with mental health.

But eventually even Republican Senate leader Mitch McConnell ended up embracing Mr. Walker as the party鈥檚 best chance to oust Democratic U.S. Sen. Raphael Warnock. The bet is that voters won鈥檛 care as much about scandals in post-Trump America.

That theory certainly got a boost Tuesday in Texas. Republican Attorney General Ken Paxton was indicted in 2015 on securities fraud charges and is still awaiting trial. He鈥檚 under investigation for corruption by the FBI and by the state bar of Texas for his role in trying to overturn the 2020 presidential election. Yet he easily won his primary against Land Commissioner George P. Bush, powered by his ability to use his office to cater to conservative causes by, for example, investigating the parents of transgender children.

Back in Georgia, firebrand Rep. Marjorie Taylor Greene won her Republican primary, shrugging off challengers who complained Ms. Greene was giving the party a bad name by engaging in Holocaust denial and other headline-grabbing, bombastic behavior.

Mr. Trump set the model in 2016, and his followers are perfecting it聽鈥� never let a scandal get in the way of winning an election.

A game of inches

The marquee Democratic matchup of the night聽鈥� in Texas鈥� 28th Congressional District, between progressive Jessica Cisneros and centrist Rep. Henry Cuellar, one of the last abortion rights opponents in the Democratic caucus聽鈥� was too early to call Wednesday.

The two candidates were separated by the thinnest margin of votes in a rematch two months after they were forced into a runoff. It drove home two realities: Elections are a game of inches, and even a victory won鈥檛 resolve the great left versus center divide in the Democratic Party.

After the collapse of much of Mr. Biden鈥檚 agenda in Congress, progressives have gotten a boost in recent primaries. Their candidate, Summer Lee, narrowly won the primary in Pennsylvania鈥檚 12th Congressional District last week. In Oregon鈥檚 5th Congressional District, centrist Rep. Kurt Schrader was trailing a progressive challenger after their primary last week; the results were delayed by ballot counting problems.

Also Tuesday, Rep. Lucy McBath handily defeated Rep. Carolyn Bourdeaux in the Democratic primary in Georgia鈥檚 7th Congressional District in the Atlanta suburbs. While neither has embraced the left wing of the party, Ms. Bourdeaux was better known as a moderate than Ms. McBath.

Still, the left lost a key congressional primary in the Cleveland area just a few weeks ago. They had an awful track record in 2020. And some Democrats worry聽鈥� and Republicans hope聽鈥� that leftist wins in places like Oregon鈥檚 5th or Texas鈥� 28th will make it harder for the party to hold those relatively moderate districts, especially in what鈥檚 looking like a dismal fall for Democrats.

Sometimes, though, races are so close that there鈥檚 eventually a winner but no resolution to the political debate they embody. Progressives can note Ms. Cisneros improved her margin after losing to Mr. Cuellar in 2020. Centrists can point to how the incumbent kept it close even amid the party鈥檚 fury about a possible end to the right to abortion.

After Texas, the struggle between the Democrats鈥� left and centrist wings seems poised to go on.

This story was reported by The Associated Press.