Miami (CNN) The oppression felt by many under the Castro regime still looms large at the epicenter of Cuban America: Miami's Little Havana neighborhood.

Cuban-born Americans at the popular Domino Park chat casually in Spanish as the domino tiles click, but most of them refuse to discuss President Barack Obama's new policy of normalizing relations with the communist country.

This past weekend, Obama shared a historic handshake with Raul Castro at the Summit of the Americas, and on Tuesday, Obama told Congress he intended to take Cuba off the state sponsors of terrorism list, moves that would have been unthinkable just years ago.

The Cuban immigrants in Miami, though, are concerned that if they talk about Obama's actions, their words could travel back to Havana and cause trouble for their families.

It's a fear, said Helena Jimenez, that's very real: "You talk the wrong way, you're put in jail, your family is hurt."

And yet Jimenez, who immigrated to the United States from Cuba when she was 8 years old, nearly 55 years ago, supports Obama's new policy.

Photos: Summit of the Americas Photos: Summit of the Americas President Barack Obama holds a joint press meeting with Cuban President Raul Castro at the VII Summit of the Americas in Panama City, Panama, on Saturday, April 11. Regional leaders gathered for the historic summit that saw U.S. and Cuban leaders sit face to face for the first time in more than 50 years. In December, Obama announced he was seeking to renew diplomatic ties with Cuba after half a century of strife, including eventually opening embassies in Washington and Havana. Hide Caption 1 of 8 Photos: Summit of the Americas Obama joins other world leaders for the opening plenary of the Summit of the Americas on April 11. Castro, center, and Cuban Foreign Minister Bruno Rodriguez, left, are among those listening to Obama's remarks. Hide Caption 2 of 8 Photos: Summit of the Americas Obama addresses the opening plenary of the Summit of the Americas on April 11. Susan Rice, the U.S. national security adviser, is at right. Hide Caption 3 of 8 Photos: Summit of the Americas Presidents and heads of state get together for a group photo at the summit on April 11. Hide Caption 4 of 8 Photos: Summit of the Americas In Havana, a couple read newspaper coverage of the historic meeting at the summit on April 11. Hide Caption 5 of 8 Photos: Summit of the Americas Obama greets Castro briefly on Friday, April 10, at a dinner for Latin American leaders at the Summit of the Americas. It was a handshake that shook the Western Hemisphere Hide Caption 6 of 8 Photos: Summit of the Americas Obama, right middle row, and Castro, left middle row, and other world leaders participate in the Summit of the Americas on April 10. Hide Caption 7 of 8 Photos: Summit of the Americas Obama holds a bilateral meeting with Panamanian President Juan Carlos Varela in Panama City on April 10. Hide Caption 8 of 8

Speaking outside Café Versailles, the famous Cuban eatery where exiles young and old gather for café con leche, Jimenez called the U.S. embargo against Cuba a "cane of support" that Cuban President Raul Castro has been leaning on as an excuse for poor conditions in his country -- and now that the cane is gone, she's hopeful he could tumble and fall.

Jimenez is emblematic of a broader shift within in the Cuban population in America, one that ultimately opened up the political space for Obama to start to change the U.S. stance toward the island nation.

But the same shift in attitudes that helped pave the way for Obama's course change has also diluted some of the potency of what was once a clear political winner for Republicans. Fernand Amandi, a principal at Bendixen & Amandi International, which specializes in polling Cuban Americans, said what was once a reliable GOP base vote has now become more of a swing vote.

"Going into 2016, the Cuban American vote in Florida will arguably be even more important than ever before, because now you're going to see both sides, Republicans and Democrats, go after them in ways we haven't seen before," he said.

Rubio faces roadblocks with Cuba policy

It's a transformation that makes Florida GOP Sen. Marco Rubio's intense opposition to the Cuba opening a potential obstacle in his presidential pitch to voters who share his background, rather than the surefire crowd-pleaser it has long been.

The senator leaned heavily on the story of his immigrant parents' exile from Cuba in his presidential announcement on Monday, and he chose Miami's Freedom Tower, which served as a landing hub for many Cuban exiles through the latter half of the 20th century, for the launch itself.

Though the 43-year-old cast himself as a leader of the future, his focus on reasserting tough sanctions and freezing relations with Cuba is seen by many, especially younger Cuban Americans, as a policy of the past.

And that was the gist of the attack launched by the Democratic National Committee when Rubio on Tuesday condemned Obama's call for Cuba to be removed from the state sponsors of terrorism list.

"For a guy who just yesterday said he wanted to be a new leader and usher in a new American century, it sure sounds like Marco Rubio is clinging to an outdated foreign policy relic from the Cold War," said DNC spokesman Mo Elleithee.

Distance and time have transformed the Cuban American population, and in turn, its politics.

Surveys suggest younger Cubans are more open to liberalizing trade and normalizing relations with the nation. A Florida International University poll in 2014 found that majorities of Miami-area Cuban Americans supported unrestricted travel to the nation, expanded trade opportunities and, broadly, stronger ties between the two countries.

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Those majorities declined among older generations, and in many cases reversed to majorities opposed among respondents that left Cuba in the 1960s and '70s. But the Cuban American population is becoming younger and more likely to have been born here -- those born in Cuba declined from 68% in 2000 to 57% in 2013, according to the Pew Research Center

Orlando Feliciano, a 20-year-old sophomore at Miami University, is one of those young Cubans who say it's time for change. He described his family as split along generational lines, with his mother in favor of the move and his grandparents opposed "because they feel like, that's a strategy that the Cuban government is using to generate more money."

But Feliciano's own view is that thawing relations is the right decision.

"I mean, Cuba has been stuck with this government for over 50 years now, and I feel like it's time for something to change," he said Monday on his way to class on the sunny campus. "The people deserve it."

Carl Meacham, director of the Americas Program at the Washington-based Center for Strategic and International Studies, said it was difficult to "pin down" the Cuban American population as a voting bloc because of the diverse opinions within it.

Cuban American voters split

"They are much more split today on the way forward than they have ever been," he said. "You have more Cuban Americans supporting a change to how the United States is supporting Cuba than you have in the past."

But it's still a potent political issue for older Cuban Americans, which helps explain Rubio's -- and the rest of the GOP field's -- Cuba focus, as older voters remain a key portion of the GOP base.

An MSNBC/Telemundo/Marist poll out last week found a majority of Americans, spanning all age groups, approved of the move to reopen diplomatic ties with Cuba, but about a third of those older than 45 disapproved.

And Meacham noted that the stance on Cuba plays into the broader GOP campaign against executive overreach from the President, with Republicans portraying it as yet another example of Obama acting without congressional approval.

Part of the policy shift stems from Obama being a second-term president unconcerned with the politics of the issue -- and perhaps wanting to test his own theories of international engagement.

That's what Jaime Suchlicki, director of the Institute for Cuban and Cuban-American Studies at the University of Miami and an opponent of the thaw, said was behind Obama's move.

"He comes with a philosophy based on the idea that you make nice with your enemies and they become your friends," Suchlicki said. "The President is doing this based on his own mindset, not on the reality of what is happening."

He warned: "The administration takes a significant risk that by taking this unilateral action, they get nothing from Cuba."

Potential political backlash from the estimated 2 million Cuban Americans in the United States -- a huge portion of whom live in the key swing state of Florida, with a sizable enclave in heavily Democratic New Jersey -- has helped keep previous Democratic presidents from moving on promises to normalize relations.

JUST WATCHED Obama-Castro meeting begins new era Replay More Videos ... MUST WATCH Obama-Castro meeting begins new era 02:59

That was the case in the early 1990s, when President Bill Clinton ended up signing the Helms-Burton Act, which further entrenched the embargo and restricted the president's ability to roll it back, rather than moving toward normalization.

"[It] was good election-year politics in Florida," he admitted in his memoir, before acknowledging that "it undermined whatever chance I might have if I won a second term to lift the embargo in return for positive changes in Cuba."

Meacham, who supports the policy shift, said the fact that Obama was freed from some of those concerns likely influenced the move.

"The President isn't going to have to run for office again and wasn't being overly careful with maintaining support in places like Florida, where the Cuban vote is pivotal," he said.

But not only do Democrats have less reason to fear a backlash on the Cuba shift, they have more Cuban voters to count on their side of the aisle.

Breaking toward the Democratic Party

Cuban Americans historically have been seen as a monolithic Republican voting bloc, but polling shows they have started to switch allegiances as the population has grown younger and more distant from Cuba.

June 2014 Pew survey indicated that registered voters of Cuban descent were about evenly split down party lines, with 42% identifying as Democrats and 47% identifying as Republicans -- down from 64% who were registered Republican in 2002.

Those shifts, Meacham argued, have made the Cuba issue "one in which the train has sort of left the station."

And they also mean that Rubio could have even less to gain by making the Cuba embargo a major issue.

"[Republicans] run the risk of losing a big chunk of support within the Cuban community" by campaigning so aggressively against the diplomatic thaw, Amandi warned. "It's not the type of issue where they can count on appealing to all Cuban Americans anymore."

It is, however, the issue that entices 73-year-old Miguel Coppola, who came to the United States from Cuba on New Years' Eve 1951, to Rubio. After finishing a cafecito at the Versailles, Coppola said that "of course" the Florida senator will have the support of the Cuban American community.

"He's got the same values we have," he said, listing "freedom in Cuba, freedom of speech, electing people in Cuba against dictatorships" as examples of those values.

But for others, particularly the younger immigrants, it makes less of a difference.

Jimenez, who left Cuba in the late 1960s, said the Cuba issue isn't much of a deal-breaker. She said she's most interested in former Florida Gov. Jeb Bush, because "what we need in this country is a good person."

And anyway, she said, "It'll be hard to go back to the way we were," with an embargo against Cuba.

"It hasn't worked for more than 50 years, so why go back to that, you know?" she said.