I can only guess since I missed about 20 years of automotive progress but let me say this. The vehicles in my day had an automatic choke that was a spring that would change as it heated up and adjust the throttle on the engine for the correct air/fuel mixture. Todays vehicles seem to know the engine temperature and adjust the length of time that the injectors fire in order to maintain the correct Air/Fuel ratio.
It sounds like your Air/Fuel ratio may be incorrect when the engine warms up or some other component starts to fail when it heats up.
Another problem that can occur is vapor lock. The gas in the gas line can get so hot that it changes from a liquid to a gas.