I have a 2.5L four-cylinder that hesitates and stumbles badly if you stop the engine for a short time (3 to 10 minutes). If you stop for less that 3 minutes or more than 10, it runs fine when you subsequently start it. I never experience this problem after a cold start, and only sometimes experience it after a hot start as described. The stumbling and hesitation ceases after the car has again run for about 5 minutes.
If the engine has a carburator I would start with looking at the choke and see if something is going on there.
Sorry, the 2.5L has single-point fuel injection. Any other ideas for me to chase down?