Those Cubans are now joined by Venezuelans fleeing communism.
The way to counter leftist appeal among any group, race, or region is to do what the leftists do. Organize!!
They have not cornered the market on ideas. As a matter of fact, their ideas suck and ours work. See Ronald Reagan. See the ever more successful president, Donald J. Trump.
Working American ideas can appeal to anyone, and will.
If Reagan ran today for anything in California he would finish a distant third behind the top two democrats. The population that elected him statewide four times has been replaced by a new population, one that is hostile to the United States.
Millions of dollars has been spent trying to win over this new population to the ideas of liberty, free-markets, and equality under the law. It has ended in ashes.