There is also something called "Pursuiter's Cough" and I am not 100% sure, but I think it is different from "Exercise-induced Asthma".

I typically get Pursuiters Cough after a time trial event. Sometimes I will get it after a very hard road race. I don't get it after training rides.

In a timetrial you are breathing so hard and heavy, it is almost as if the dryness of your throat combined with the heavy breathing "sand-papers" the back of your throat. Consequently, I often have a dry (occasional) cough which can last a couple of hours, or sometimes a couple of days.

When I don't cough, I take it as a sign that I didn't push myself hard enough!