Reviewing Mariel Boatlift.

General history

Over 2,000 entrants were jailed or detained upon arriving in Florida during the exodus. Some were deported back to Cuba, but many remained imprisoned for several years after the boatlift ended.

https://www.floridamemory.com/blog/2017/10/05/the-mariel-boatlift-of-1980/

An interesting part is

https://en.wikipedia.org/wiki/Refugee_Act

The main objectives of the act were to create a new definition of refugee based on the one created at the UN Convention and Protocol on the Status of Refugees, raise the limitation from 17,400 to 50,000 refugees admitted each fiscal year, provide emergency procedures for when that number exceeds 50,000, and to establish the Office of U.S. Coordinator for Refugee Affairs and the Office of Refugee Resettlement.

The annual admission of refugees is set to a 50,000 cap per fiscal year, but in an emergency situation, the President may change the number for a period of twelve months. The Attorney General is also granted power to admit additional refugees and grant asylum to current aliens, but all admissions must be reported to Congress and be limited to 5,000 people.

This link leads to a historical chart of refugees http://www.wrapsnet.org/admissions-and-arrivals/