B.B.
:-) You don't "get" asthma... You always have it... But it can be a nonissue for some until the right situation irritates it. Not every one who has asthma has the full blown attacks that others have... I have an inhaler to be used as needed... I only need it when I get a cold... Other wise the coughing lingers for months. But the inhaler helps settle the irritated lungs and I can sleep and breathe better...
From your description it sounds more like he's just clearing his lungs out and then inhaling to refill them fully...
If it keeps up then mention it to the dr at your next appt. Could be that he just got a bit of a cough and phlem build up that he notices when he's moving around but once he clears it out doesn't bother him till he's inactive again for a bit... Do you notice it more right after or during really active play... Or is it more of him playing on a carpet (stiring up dust) or petting the dog?...