In kid's your son's age, shortness of breath should be looked at closely. You have taken the right first steps to have you son evaluated for asthma, which is one of the more common causes of cough and shortness of breath. Make sure that your son has had a full evaluation including pulmonary function testing with bronchdilatory challenge (needed for asthma diagnosis). A simple history of his symptoms may not be sufficient to rule asthma out. Cough and shortness of breath can also occur because of bronchitis. If your son's symptoms occur during times when he was otherwise not feeling well this could explain his symptoms. Your son should be evaluated for vocal cord dysfunction. This occurs when the vocal cords paradoxically shut instead of open when the person breaths in. The symptoms that occur is difficulty getting a breath in, and subsequent shortness of breath. It is often mistaken for asthma. Although it is rare in young people, these symptoms could also be heart failure. This, obviously would be very unusual.
I suggest that you schedule an appointment with your son's pediatrician
. You can discuss your son's work up this far, and ask what additional diagnostic tests will be done. Your son may benefit from referral to a pediatric pulmonologist