My oldest son, at 14 mos., was hospitalized for troubled breathing...that was just the beginning. The cough you speak of, it's the kind that you just want to scream, "enough already!" Dry, persistent & just goes on...
Every asthmatic is different & therefore needs to be treated as such. My son was hospitalized or had ER visits so often I lost count. Winter, dry heat, aggravated his condition & I can't tell you how many vaporizers I went through, having them running constantly during the months when the heat was on. We couldn't get a handle on it while under the pediatrician's care....not until I found a specialist (Pulmonologist, now retired) could he begin to stay away from hospitals. On my first visit to him, he sent me home with the "smoke blower machine", as my son called it, for at home breathing treatments.
You're doing the right thing by getting to a specialist. My son was affected, severely, with any virus/cold, suffered from allergies (which compounded his condition) & exercise induced asthma (being excited, running around, etc).
Meds now are amazing to control & maintain this condition. My advice, don't treat your son as though he's sick....deal with the attack, meds, doctors & move on...it's something he'll most likely have to deal with for his whole life & none of us wants to instill a "your sick, kid" attitude on any of our kids! :-) Good luck!!