The idea behind putting them outside is that you're not keeping them in a moldy barn where there is lots of hay, dirt, animal dander, etc. concentrated. It is all about reducing irritation to the airway and though you're not eliminating allergens by putting them outside, you are at least alleviating some of the contact.
As far as management goes the standard treatment is corticosteroids (such as dexamethasone) to reduce inflammation and bronchodilators (such as albuterol) to help open the airways. No you are not addressing the specific antigen (allergen) that is creating the problem, you are addressing the body's response to it. HTH.