Immigration a top issue for Republican governor candidates ... but what can they actually do about it?

Immigration came up more than education, transportation, job creation and the environment -- combined -- during last month's Republican gubernatorial debate.

Both Agriculture Commissioner Adam Putnam and U.S. Rep. Ron DeSantis have vowed to stand with President Donald Trump on his border policies, even when the separation of migrant families came under bipartisan criticism.

A plurality of Republicans list immigration as the No. 1 issue facing the country, according to a June Gallup poll, even though -- despite an uptick in recent months -- apprehensions at the border are historically low compared to the previous two decades. Trump himself has pointed this out.

Still, the issue dominates the campaign trail for a governor's job that doesn't have obvious authority over immigration.

"A governor is pretty much toothless," said Arturo Rios, an immigration law professor at Stetson University. "As far as changing major policy and forcing immigration statutes, the governor has essentially no power."

The U.S. Supreme Court reaffirmed the federal government's authority to enforce the country's border and immigration laws in 2012 when it gutted Arizona's attempt to circumvent President Barack Obama's immigration policies. The state's law, deemed invalid, banned undocumented workers, imposed stiffer penalties on unlawful residents, and allowed warrantless arrests of anyone suspected of committing a deportable crime.

But since that decision, states like Florida have found other ways to address undocumented populations.

When Rick Scott became governor in 2011, he vowed he would implement an Arizona-style immigration law, a promise PolitiFact later rated "Broken."

Scott instead moved to require state agencies and contractors to use E-Verify, a computer program that checks whether a prospective employee is a legal resident.

E-Verify remains a contentious issue in Florida among Republicans, where the GOP is split between anti-illegal immigration hardliners and those representing farms and other businesses that rely on migrant labor.

That tension split June's GOP debate. DeSantis accused Putnam of maneuvering behind the scenes on behalf of the agriculture industry to kill a bill that would have required private businesses to use E-Verify before hiring.

"He put his big donors who want cheap foreign labor ahead of the interests of our citizens," DeSantis said.

Putnam didn't deny the charge, and instead repeated a debunked attack that DeSantis voted on an agricultural bill "to give food stamps to illegal immigrants." Putnam's campaign this week declined to say whether he supports E-Verify.

DeSantis has criticized Putnam's Congressional record in support of an eventual path to citizenship for some undocumented migrants, calling him "Amnesty Adam."

In two TV ads featuring Florida sheriffs, Putnam has pushed back against DeSantis' line of attack by promising to support Trump's border policies and help the administration deport undocumented immigrants who have been charged with crimes. As news broke that Trump's policies split families and put thousands of children in migrant camps, including one in Homestead, Putnam sent out a fundraising e-mail declaring, "I'm proud to support the President's agenda."

On issues like amnesty, border security and deportation, a governor's only real power is the bully pulpit, said Rios, the immigration attorney.

But their ability to influence Washington, D.C., through TV appearances or strongly worded letters can be overstated, said Adam Basford, a lobbyist for the Florida Farm Bureau.

"That's why we elect Senators, it's why we elect Congress people," Basford said. "A governor has bigger fish to fry."

Both candidates have denounced so-called sanctuary cities, where law enforcement agencies are discouraged from working with federal immigration officials, though no city in Florida has officially declared itself a "sanctuary." But they've been mum on immigration issues where as governor they could play a role.

Only California and Texas have more undocumented workers than the 850,000 who live in Florida.

Take schools. Many districts must educate migrant populations who may not speak English and have expensive challenges.

"A governor could go to the Congress and say we're getting an unexpected number of these immigrants, they present a lot of issues, and if you want us to serve them, you have to provide us the resources," said Don Gaetz, a former Republican state senator and school superintendent in Okaloosa County. "Or a governor can say to his own state, 'We need to do more to help.'?"

During a different political era, Gov. Jeb Bush wrestled with the Legislature in 2004 over whether undocumented workers should be eligible for a driver's licenses if they pay car insurance. They still can't, however, though many drive unlicensed and uninsured.

As recently as 2014, Scott signed a law that allows some children of undocumented aliens to pay in-state tuition to attend Florida universities. Lawmakers have since attempted to undo the legislation.

When asked about it this week, campaigns for Putnam and DeSantis wouldn't weigh in on either measure.

Contact Steve Contorno at scontorno@tampabay.com. Follow @scontorno.