AUSTIN, Texas — Gov. Ron DeSantis of Florida wanted to irritate a set of wealthy, liberal elites when he flew migrants to Martha’s Vineyard from Texas, delivering them a slice of the humanitarian crisis simmering along the nation’s southern border.

But Mr. DeSantis’s stunt also annoyed an entirely different group — fellow Republicans in Austin, including some of the allies and aides of Gov. Greg Abbott of Texas.

Publicly, Mr. Abbott has not criticized Mr. DeSantis’s migrant flights from his state. “Every state that wants to help, I’m happy for it,” said Dave Carney, Mr. Abbott’s top campaign strategist.

But privately, the Florida governor’s gambit stung Mr. Abbott’s team. No one in the Texas governor’s office was given a heads-up that Mr. DeSantis planned to round up migrants in San Antonio, according to people familiar with the matter.

Mr. Abbott had spent months — and millions of state tax dollars — methodically orchestrating a relocation program that, since April, had bused 11,000 migrants to Washington, New York and Chicago. Mr. DeSantis’s adaptation was considerably smaller.

But it immediately put the national spotlight on Mr. DeSantis, garnering headlines and earning him praise from Republicans and condemnation from Democrats. It also led to an investigation by the sheriff in San Antonio and a lawsuit from migrants who said they had been lured onto the planes under false pretenses. Mr. DeSantis grabbed the attention of right-wing America, using Mr. Abbott’s tactic, on Mr. Abbott’s turf, to bigger and more dramatic effect.

Mr. DeSantis’s instinct for political theater has helped him quickly turn into Republicans’ leading alternative to former President Donald J. Trump. Even Texas Republicans tell pollsters that they prefer Mr. DeSantis over Mr. Abbott for president in 2024.

The two Republican governors have been locked in an increasingly high-stakes contest of one-upmanship, wielding their own unique brands of conservatism and pushing boundaries by using desperate migrants for political gain. In Florida, Mr. DeSantis mused to donors last year about Mr. Abbott’s good political fortune to share 1,254 miles of border with Mexico and complained that he didn’t have the same to use as a backdrop, according to one person familiar with the conversation.

For all the bluster, the war between Austin and Tallahassee is decidedly more cold than hot. Yet, the two governors’ policy moves antagonizing the Biden administration and the Democratic Party as a whole have been unfolding as an interstate call and response, with national repercussions.

In August 2020, Mr. Abbott proposed legislation to punish cities in Texas that took funding away from police departments by preventing them from raising more property tax revenue. The following month, Mr. DeSantis went further, saying he would seek to cut state funding from municipalities that defunded the police.

In February of this year, Mr. Abbott ordered state officials to open child-abuse investigations into medically accepted treatments for transgender youth, including hormones and puberty-suppressing drugs. Last month, Mr. DeSantis said doctors who “disfigure” young people with gender-affirming care should be sued.

In June 2021, on the first day of Pride Month, Mr. DeSantis signed a bill into law that barred transgender girls from playing on female sports teams at public schools. Mr. Abbott followed suit with a similar measure that October.

The competition between Mr. DeSantis and Mr. Abbott has more to do with their job descriptions than any personal animosity. Governors elected to lead megastates like Florida and Texas — two of America’s three largest states that accounted for 15 percent of the Republican presidential vote in 2020 — are automatically injected into the national political arena, where they are sized up and watched closely for signs of White House ambitions.

“Love Florida. Love Texas. Love Florida more,” Jeb Bush, a former Florida governor with deep familial ties to Texas, said when asked about the rivalry between the two states.

When Rick Scott was the governor of Florida and Rick Perry was the governor of Texas, the two Ricks shared a bromance even as both eyed the White House. From Florida, Mr. Scott spoke glowingly of his counterpart’s record of luring businesses. In Texas, Mr. Perry admired his rival’s refusal to accept federal stimulus money for railroads or to expand Medicaid.

Mr. DeSantis and Mr. Abbott have lacked such camaraderie. Their brinkmanship has played out against the backdrop of their re-election bids. Both men are seeking additional four-year terms while facing challenges by well-known Democrats in contests that could help determine their presidential aspirations and the direction of the Republican Party for years to come.

“No one has ever been elected governor of even a small state who didn’t, somewhere deep in their heart, start dreaming about being president,” said Chris Wilson, a pollster who has worked for both men. “So it’s not shocking to see both Abbott and DeSantis jockeying at least a little toward 2024 or beyond.”

Mr. Abbott is the more institutional politician.

He faced no opposition in his first primary election for attorney general in 2002, and was effectively unopposed inside the party when he ran to succeed Mr. Perry as governor in 2014. He has worked to maintain ties with business groups, social conservatives and fellow Republican governors. A former Texas Supreme Court justice, he is a rather lawyerly governor.

Mr. DeSantis is more instinctual.

He emerged from a six-way Republican primary in his first race for the House of Representatives in 2012. He was viewed as an underdog in the 2018 governor’s primary until he became separated from the pack, thanks to an endorsement — and constant promotion — from Mr. Trump. A former lawyer for the Navy at Guantánamo Bay, he is more pugilistic than judicial.

Still, Mr. DeSantis has positioned himself as something of a political loner.

He has eschewed events coordinated by the tight-knit Republican Governors Association. Instead of joining a group of current and former Republican governors on the campaign trail this year to support fellow incumbents, Mr. DeSantis embarked on his own victory lap, promoting the migrant flights during campaign stops with Republican candidates for governor in Kansas and Wisconsin. Those events were organized not by the Republican Governors Association, but by Turning Point USA, a group of younger and more provocative conservative activists close to Mr. Trump and his family.

In Tallahassee, the migrant flights had been discussed for more than a year and had, at one point, centered on relocating the migrants to the Hamptons, the popular Long Island destination for wealthy New Yorkers, according to people familiar with the talks. Initially, the proposal caused some division within Mr. DeSantis’s team.

The contrasting styles of Mr. Abbott and Mr. DeSantis were on clear display last year in their handling of high-profile election bills.

When Mr. Abbott signed a new round of voting restrictions, he traveled to Tyler, the hometown of one of the bill’s chief proponents, and was surrounded by Republican lawmakers, supporters and reporters. When Mr. DeSantis signed his state’s voting restrictions, he, too, was surrounded by fellow Florida Republicans, but the only network that was allowed to cover the event was Fox News, which aired the footage live on its program “Fox & Friends.”

The coronavirus pandemic has been a defining moment for both governors.

Mr. DeSantis burnished his conservative bona fides by challenging Covid safety guidelines from public health officials. He lifted pandemic restrictions on businesses in Florida in September 2020, earlier than most governors.

By contrast, Mr. Abbott found himself clashing with conservatives over the business restrictions and mask mandate that he had ordered. Some donors confronted Mr. Abbott, expressing their disappointment that he was not following Mr. DeSantis’s lead and suggesting that he could lose re-election if he did not move quicker to reopen businesses and return the state to normalcy, according to two Republicans who participated in the meeting.

Mr. Abbott eventually lifted restrictions on businesses in March 2021, months after Mr. DeSantis did.

“Governor Abbott and Governor DeSantis have a solid working relationship, having worked together on various initiatives through Republican governors organizations,” Renae Eze, Mr. Abbott’s press secretary, said.

A spokeswoman for Mr. DeSantis did not respond to requests for comment about his relationship with Mr. Abbott and his remark about the border to donors last year.

Last year, after the start of the Biden administration and as migrants arrived at the border in increasing numbers, Mr. Abbott and Gov. Doug Ducey of Arizona sought assistance from other states to help police the border.

Florida sent a large contingent of officers and equipment, including low-water boats that could be used along the Rio Grande in Texas.

Soon after, Mr. DeSantis planned a trip to visit the Florida officers stationed at the border. When members of Mr. Abbott’s office learned about the Florida governor’s trip, they timed a news conference to Mr. DeSantis’s arrival and invited him, according to a person familiar with the plans.

On the day of the news conference in July 2021, in an airplane hangar in Del Rio, Texas, Mr. DeSantis arrived long before Mr. Abbott. He met with Florida officers and officers from Texas, and then spent an extended period of time sitting on a couch in the hangar waiting for Mr. Abbott, who arrived about 10 minutes before the briefing.

The two men were dressed in similar black button-down shirts with their respective state seals embossed over the left breast pocket (long sleeves for Mr. DeSantis, short sleeves for Mr. Abbott). They took their positions at a table set up in front of a boat, a plane and a helicopter that had been used to patrol the border. Then they proceeded to stumble over one another at the start of the news conference.

“I think we’re ready to go,” Mr. Abbott began.

“OK,” Mr. DeSantis said, looking into the cameras.

“Good,” Mr. Abbott added.

“Well,” Mr. DeSantis said, his right elbow pointing toward the Texas governor seated next to him.

“So,” said Mr. Abbott, looking back at Mr. DeSantis. “Oh. Help kick it off?”

“I just want to thank ——” Mr. DeSantis said.

“Sure,” Mr. Abbott said.

“Thank you, Greg, for hosting us,” Mr. DeSantis said, turning back to the cameras.

Mr. Abbott invited fellow Republican governors to another news conference at the border that October. Ten showed up.

Mr. DeSantis was not among them.