Gov. Ron DeSantis of Florida on Friday swept through South Carolina, his rival Nikki Haley’s backyard, seeking to blunt the rise of the state’s former governor in the Republican primary while capitalizing on his slugfest with the Democratic governor of California.

The fight to claim the mantle of the most viable Republican alternative to former President Donald J. Trump intensified this week. Ms. Haley, a former ambassador to the United Nations, won the endorsement of the political network founded by the billionaire industrialist brothers Charles and David Koch and secured the backing of key donors like Jamie Dimon, the chief executive of JPMorgan Chase. The upheaval in Mr. DeSantis’s campaign apparatus continued unabated on Friday with the departure of his super PAC’s chairman, Adam Laxalt.

But Mr. DeSantis’s prime-time face-off on Fox News with Gov. Gavin Newsom on Thursday night seemed to buoy him as he barnstormed through South Carolina’s Upstate region near Greenville and its midsection outside Columbia before ending in Charleston and the Lowcountry.

On Ms. Haley’s home turf, the Florida governor and his surrogates appeared unfazed by all the developments, blasting the support from Mr. Dimon as a nod from a Hillary Clinton supporter and dismissing the Koch network’s decision as evidence that Ms. Haley represents only the incremental change backed by Wall Street.

“I don’t know what you could say about her tenure as governor here, but I’ve never heard of any major accomplishment that she had,” Mr. DeSantis told a town hall in Greer, S.C. “And I don’t think she’s shown a willingness to fight for you when it’s tough. It’s easy when the wind’s at your back.”

Mr. DeSantis and his team have long cast the Republican nominating contest as a two-man race between him and Mr. Trump. But Ms. Haley’s rise in the polls and her successful drawing in of big-money donors have punctured that notion. After ignoring each other for much of the summer and the early fall, the two candidates now engage in near-daily attacks and have sparred with increasing intensity on the debate stage.

They and their allied super PACs have clashed on a sometimes bewildering variety of fronts, from standard disputes over subjects like the strength of their conservative bona fides to more niche topics like dissecting each other’s records on recruiting Chinese businesses in their home states. Fact checkers have rated many of their attacks as false or misleading.

Mr. DeSantis has been especially aggressive. On Fox News this week, he called Ms. Haley an “establishment” politician who was “fundamentally out of step with Republican voters” on core conservative issues, including immigration, where she has called for more legal pathways to recruit foreign workers. His campaign set up a website that accuses Ms. Haley of supporting “every liberal cause under the sun.”

The Florida governor has also falsely claimed that Ms. Haley wanted to bring Gazan refugees to the United States. And he has mounted attacks on past statements by Ms. Haley that have sometimes mirrored his own, including her expression of sympathy after the murder of George Floyd in 2020. (Mr. DeSantis said at the time that he was “appalled” by Mr. Floyd’s death.) He has also criticized Ms. Haley for saying in 2016 that her state did not need a law banning transgender people from using the bathrooms of their choice, even though Mr. DeSantis had said during his first run for governor in 2018 that “getting into bathroom wars” was not “a good use of our time.”

In addition, Mr. DeSantis’s allies have established a new super PAC to attack Ms. Haley in Iowa. Last month, the group, Fight Right, produced a misleading ad showing clips of Ms. Haley praising Hillary Clinton. The clips had been edited to excise her simultaneous criticisms of Mrs. Clinton.

The Haley campaign counterpunched with an ad of its own called “Desperate Campaigns Do Desperate Things.”

As Ms. Haley and Mr. DeSantis joust for second place, Mr. Trump’s position in the Republican primaries, even in South Carolina, remains dominant. Mr. DeSantis may have been willing to call out Ms. Haley by name in her home state, but he continued to tiptoe around the former president using the passive voice.

On Friday, for instance, he insisted he could get Mexico to pay for the rest of a border wall.

“I know that was promised — it didn’t happen,” he said, without naming the promiser, Mr. Trump.

Mr. DeSantis also failed to identify that same promiser when he said: “We were promised we’d be tired of winning. Unfortunately, as a Republican, I’m tired of losing.”

He was less reluctant when a voter asked about Ms. Haley’s suggestion last month that social media platforms verify all users and ban people from posting anonymously. Mr. DeSantis and others had criticized her comments as unconstitutional and a threat to free speech.

“What Haley said was outrageous,” Mr. DeSantis responded, adding, “Honestly, it’s disqualifying.”

In an interview on Fox News, Ms. Haley responded to Mr. DeSantis’s recent critiques about her record as governor.

“Well, I think he went after my record as governor because he’s losing,” Ms Haley said. “I mean, who else can spend a hundred million dollars and drop half in the polls?”

For some in South Carolina, Ms. Haley’s tenure — from 2011 to 2017 — is beginning to feel like a long time ago. Tim Vath, 54, who moved to Greer toward the end of Ms. Haley’s second term, found the support from the Koch network “questionable.” Suzanne Garrison, also 54 and from Greer, raised the specter of backroom politicking and suggested that Ms. Haley had not always followed through in implementing conservative policies.

“I wish more people knew the real Nikki Haley,” said Ms. Garrison, a DeSantis supporter. “I just don’t trust her.”

But in the tiny town of Prosperity, outside Columbia, the crowd was more mixed. Cathy Huddle, 61, of Chapin, S.C., said Ms. Haley’s accomplishments as governor “pale in comparison” to the wholesale conservative changes wrought by Mr. DeSantis in Florida.

But Alice and Robert Tenny appeared unmoved by Mr. DeSantis’s pitch. Both said Ms. Haley’s experience at the United Nations gave her global knowledge and stature. And Mr. Tenny, 69, found Mr. DeSantis’s gloating over his performance against Mr. Newsom to be off-putting.

“We’re kind of getting ahead of ourselves if we’re sitting down with the governor of a different party” before the first primaries, Mr. Tenny said.