Even in Miami, where old-guard positions remain popular among older exiles, who are largely Republicans, there have been notable changes. In 2012, Joe Garcia became the first Cuban-American Democrat from Miami to be elected to the House. While he publicly supports the embargo, Mr. Garcia holds views significantly different from other South Florida members of Congress. For instance, he has called for clinical trials in the United States of a Cuban diabetes treatment that has shown great promise. He also favors easing travel restrictions to the island.

Still, ending the embargo, which requires congressional action, remains challenging because a small but passionate group of Cuban-American lawmakers is adamant about maintaining the status quo. The most vocal defenders of the embargo are Senator Robert Menendez, a Democrat from New Jersey; Senator Marco Rubio, a Republican from Florida; and Representative Ileana Ros-Lehtinen and Representative Mario Diaz-Balart, both Miami Republicans.

In April, during the height of the crisis set off by Russia’s invasion of Crimea, Mr. Menendez, the son of Cuban immigrants who moved to the United States in 1953, delivered a long, impassioned speech on the Senate floor, arguing that despite the myriad foreign policy crises in the world, Washington needed to focus on the abuses of “a Stalinist police state” 90 miles away. He displayed photos of dissidents and warned that expanded travel by Americans to Cuba was enabling a despotic state. White House officials fear that Mr. Menendez, as the chairman of the Foreign Relations Committee, could hold up confirmation of federal nominees in retaliation for further moves to ease the embargo.

Mr. Menendez’s loathing of the Cuban government has only increased because he believes the island’s intelligence service sought to destroy his career by planting a fabricated story in the media suggesting that he had patronized underage prostitutes in the Dominican Republic.

White House officials are less concerned about pushback from Republicans, who are reflexive about criticizing the president on foreign policy. While a growing number of her congressional colleagues have traveled to Cuba, Ms. Ros-Lehtinen, who is among the most ardent supporters of the embargo, seems to be strikingly out of touch with what is happening on the island.

In a recent interview deploring a visit to Havana by Beyoncé and Jay-Z, Ms. Ros-Lehtinen expressed outrage that the celebrities had stayed in hotels where Cubans aren’t allowed to stay, even if they could afford it. As it happens, the Cuban government lifted that ban in 2008.

As the electorate has shifted on Cuba, some Cuban-American politicians have begun to call for a review of the policy that puts newly arrived Cubans on a fast track to citizenship, probably because new immigrants support closer ties with the island and grew up despising the embargo.

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Politics aside, the issue remains deeply personal for the holdouts, Cuban-Americans of that generation say, because it continues to evoke raw feelings about ancestry, homeland and loss. Those sentiments, which have lasted for more than 50 years, cannot be ignored. But they should not continue to anchor American policy on a failed course that has strained Washington’s relationship with allies in the hemisphere, prevented robust trade with the island and offered the Cuban government a justification for its failures.