You've been given some good links.
The basic problem with asthma is that the combination of swelling of the lining of the air passages and constriction of the muscles in them, narrows the channels so it is more difficult to move air, especially on exhalation. Trapping gas in the lungs while you are ascending on Scuba is a recipe for pulmonary barotrauma at best, and arterial gas embolism at worst. Therefore, diving when symptomatic from asthma is dangerous.
Asthma used to be considered a complete contraindication to scuba diving at all, but the stance on this has changed somewhat. If someone is asymptomatic on minimal or no meds, it is now considered that they can dive. If asthma is exercise-induced, however, that raises the stakes.
Each asthmatic must be considered as an independent case, and data about frequency and severity of symptoms, triggering factors, and response to medications have to be reviewed to make good decisions.
I don't know how old your child is, but one thing that would worry me a lot is that kids easily get focused on the immediate goal, and lose sight of the long term issues. I think at least some children would be quite capable of saying, "Oh, I'm FINE," when they weren't, just to get to do a desired dive.