If your son has asthma, which it sounds like he does, it will not go away. It will periodically rear it's ugly head, most likely during illnesses it sounds. You say asthma like, but it sounds like he has asthma. You will need to treat him for it. If you don't already see an allergist i would highly recommend a pediatric one.
I too have to disagree with PS about asthma being constant breathing problems. I have severe asthma and have hospitalized many times throughout my life starting as an infant, and my last hospitalization was last year and i am 40 years old now. Despite having severe asthma, I do not have, nor have I ever had constant breathing problems. Now my asthma is severe, and I have never in my life ever been able to run, not more than a few feet, without an attack. I also can't stay in the cold, and every time I get sick my asthma acts up as well as when I am exposed to anything I am allergic to. It is called illness induced asthma, exercise induced asthma and weather induced asthma, and allergy induced asthma. I have them all. But basically I just have asthma. All asthma is induced from something. Allergens are most often a culprit as is illness and weather changes and exercise. Both my kids have asthma. My daughters acts up most often when she runs or exercises, but has had some problems during illnesses. My son only has problems when he gets sick with colds, but not all colds. Everyone who has asthma reacts to different things and some only react sometimes not all the times. I am unusual in that my asthma is so bad I get it for everything. So having said all that, you have two choices of treatment. You can choose to use medication on a daily basis hoping that whatever triggers it will be minimized by daily treatment, and it usually is, so if you get flareups with illness, next time you are sick your asthma won't be as bad if you are on daily treatment. So if you put your son on daily treatment these flareups will not be as bad when he is sick. I have been on intermittent daily treatment my entire life. I take myself off when I can because the meds are not that great for a persons body, but of course it can be better than some of the consequences of no treatment. My kids on the other hand are not on any daily treatment. I have nebulizers for both of them and inhalers. I choose to treat them only as needed. Depending on the severity of their symptoms that will be re-evaluated as needed. But as I said before, if your son has asthma and sounds like he does, primarily illness induced asthma, then you need to figure out if it would be beneficial to put him on daily treatment, which I am not sure is even an option as he is so young. But you will need to treat the symptoms when he has a flareup. Things you can do are try to catch it early. If you know he is getting a cold, start the neb treatments before he is struggling to breathe. Make sure you do the treatments even if he seems okay at the moment. for example you gave him a neb treatment in the morning and he has been great all day, so you skip it before bed, and then he is up all night wheezing. Then the next day even with treatments he continues to wheeze. So if you had done the treatments throughout the day and before bed the night before, he probably would do much better. The trick with asthma is recognizing the early signs. which is really hard to do with a pre-verbal child, heck with any child. I had trouble recognizing it until I was an adult. But do what you can to try to catch it early. And treat regularly. But do see a pediatric allergist, they can give you the best options for treatment.