“The closest sort of analogy would be with border towns in Mexico, who are deeply dependent on cross-border trade,” Mr. Rubio said. He recalled attending a family friend’s wedding in the Venezuelan city of Valencia when he was younger. “You can’t live in Miami without knowing people in Venezuela.”

Ernesto Ackerman, 69, a Venezuelan-American activist who came to the United States in 1989, long before Mr. Maduro and his predecessor, President Hugo Chávez, came into power, described the arrival of subsequent “tides” of Venezuelan immigrants.

“If the problem gets solved,” said Mr. Ackerman, a Republican, “a lot of people are going to go back.”

Both Republican and Democratic lawmakers insist that they back Venezuela action on principle, not as an electoral ploy. Yet Democrats saw politics at play this month when Vice President Mike Pence delivered a Venezuela policy speech in Doral — a Miami suburb known as Doralzuela — and did not invite Democratic members of Congress. Then, on Tuesday, Mr. Trump mentioned Venezuela in his State of the Union address and immediately pivoted to a campaign line that seemed to liken Venezuela’s socialist government to liberals in the United States.

“Tonight, we renew our resolve that America will never be a socialist country,” he said.

During last year’s midterm elections, Ron DeSantis, the Republican nominee for governor, frequently deployed the word “socialist” against Mr. Gillum, his opponent. The jab, dismissed by Democrats as ridiculous, appeared to stick: Mr. DeSantis won. So did Rick Scott, a Republican who was elected to the Senate after working hard to court Venezuelans, Puerto Ricans and other Latinos who typically vote for Democrats.

Florida’s Venezuelan community has a smattering of activists who are the heads of various organizations but no obvious leader — a reflection of an immigrant group still getting established, still up for grabs politically and still glued to the news from home, usually via Twitter posts or forwarded audio files on WhatsApp.

“I’ve been sleeping maybe three hours a night,” said Yusnaiberth Detraux, 44, who described spending hours online scouring for information “until my eyes hurt.”