From what I've seen MOST exercise induced asthma can be "cured" by breathing mostly through the nose. This humidifies the air as it enters your air passageways more than the mouth which leads to less drying of the throat ESPECIALLY during the winter. Dry throat tends to cause more coughing.. possibly constriction although I'm not sure about that. One of my friends swears by this as it helped eliminate his exercise induced asthma.
It's true that asthma can go away as you get in better shape.. but not always. If you're a mouth breather then switch and see if it helps.