Mr. Trump’s dismal standing with Latino voters has complicated the task. An ABC News/Washington Post poll in mid-June found that 89 percent of Hispanics held negative views of him. And this week, the chief spokesman for the Florida Republican Party, Wadi Gaitan — the son of Honduran immigrants — left his post to join Libre, citing Mr. Trump as his reason.

“Moving on gives me a great, new opportunity to continue promoting free-market solutions while avoiding efforts that support Donald Trump,” Mr. Gaitan said in a statement.

Republicans could once rely on solidly conservative Cuban-American political refugees in Miami as a crucial source of support. But it is not so simple anymore: Young Cuban arrivals are less reliably Republican; South Americans now make up a growing segment of the Latino population in the state; and Puerto Ricans, who at more than a million statewide now rival the Cuban-American population, are flocking to Orange, Osceola and Polk Counties in Central Florida — once the heart of the state’s white working-class vote.

Those demographic changes have already had consequences: George W. Bush lost to John Kerry in Orange County by fewer than 1,000 votes in 2004, but in 2008 and again in 2012, President Obama won there by about 85,000.

Still, while Puerto Ricans generally favor Democrats, they have tended to be less party-conscious than some other groups, bolstering Mr. Obama in his two Florida victories but also helping to elect Charlie Crist as governor when he was a Republican.

And in contrast to Cuban-Americans, who have long wielded power in Florida, newly arrived Puerto Rican voters often need to be reminded they can even vote.