During the Cuban Airlift thousands of refugees left Cuba for the U.S. 

From December 1965 to early 1973, under the Johnson and Nixon administrations,
twice daily "Freedom Flights" transported from Varadero Beach to Miami. The
longest airlift of political refugees, it transported 265,297 Cubans to the United
States with the help of religious and volunteer agencies. Flights were limited to immediate relatives and Cubans already in the United States with a waiting
period anywhere from one to two years.

This photo was first published on 12/2/1965 showing the arrival of 
Cuban refugees in Miami.