That influence remains. The American trade embargo on Cuba is still in force, requiring the vote of a Congress that is reluctant to remove it. And this year’s presidential campaign saw two Americans of Cuban descent — Senators Marco Rubio and Ted Cruz — run for the office.

But in the last decade, the pro-embargo, hard-line grip on the minds and votes of people in Miami and elsewhere has weakened, creating divisions among Cuban-Americans over how deeply to engage with Cuba and its people.

With Mr. Castro’s death, some hard-liners are pushing for retrenchment and hope that President-elect Donald J. Trump will crack down now that the government has lost its father figure. Others say that this is the time to flood the zone with more people, ideas and goods. Cuba’s president, Raúl Castro, has opened the window slightly to economic reform, travel and American influence. Now that his older brother is gone, they argue, he will be freer to make changes.

But the oldest and most vehement exiles — the historicos, as they are called — are dying off in large numbers. Their children, while still passionately opposed to Mr. Castro, are open to closer ties with the Cuban people as a way of stoking change. (Even Mr. Mas Santos is part of this group.) And their grandchildren know far less about Cuba and Mr. Castro; many are more intrigued than outraged. At the same time, recent arrivals, while deeply disenchanted with the Cuban government, want to see and help their relatives on the island, above all else.

Those shifts in attitude have been translated in Washington. Two years ago, President Obama surprised Cuban-Americans by announcing a series of changes. He re-established diplomatic ties and made it easier for Americans to visit and send money and goods, and also for American businesses to establish a foothold.

“We have moved from a politics of passion to a politics of realism,” said Andy Gomez, a political analyst who was a senior fellow in Cuban studies at the University of Miami. “We are going to be passionate for the next 72 hours. But the realism is that the transition has to come within the island. The leadership has to come from within the island. I don’t think anyone in South Florida thinks they will be president of Cuba, and if they do they are fools.”