When flying internationally from the US to Latin America, the use of GPS is common. What agency or agencies control the flight and how?
Airline flights are almost ALWAYS under ATC control. The entire world is divided into Flight Information Regions. ATC control is passed from one agency to another as the aircraft passes each FIR boundary.
It is not a requirement for an aircraft to have GPS equipment as there are other navigation systems which can be used for navigating airways or direct routings.
Here is an interactive map that shows each FIR and it's boundaries: ICAO FIR WORLD Map
A flight from Miami to Bogota would fly though 5 different Flight Information Regions.