"You know we have a system now for bringing people into the country, and what we should be doing is we should be bringing people who are terrific people who have terrific records of achievement, accomplishment.
You have people that have been in the system for years (waiting to immigrate to America), and it's very unfair when people who just walk across the border, and you have other people that do it legally."
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In January, Republican presidential candidate Sen. Marco Rubio, who is of Cuban decent, filed legislation to restrict federal benefits to Cuban immigrants unless the new arrivals could prove persecution.
Unlike Trump, Rubio stopped short of wanting to prevent their escape from the communist island.
"It is outrageous whenever the American people's generosity is exploited. It is particularly outrageous when individuals who claim to be fleeing repression in Cuba are welcomed and allowed to collect federal assistance based on their plight, only to return often to the very place they claimed to be fleeing," Rubio said in a statement at the time.
Republican presidential candidate and Texas Sen. Ted Cruz, whose father was Cuban, has said he believes the Cuban Adjustment Act is still needed as long as the government there remains communist.