How does this contribute to asthma?
First, there is the "heightened alert" status. Mediators, to borrow some military lingo, are "scrambled" like fighter planes and reconnaissance planes in the clogged chambers of the upper airways -- that is, planes take off so they can either attack or target intruders for other forces. This means that the mast cells and basophils begin pumping out substances that will stimulate production of the immunoglobulins with accompanying swelling and inflammation. Some of these histamines and longer-acting leukotrienes reach the lower airways where they will look for something to attack as well. This makes the asthmatic's already "twitchy" lungs even more sensitive.
Second, there is the effect that breathing through the mouth instead of the nose has on the lungs. The nose provides a service by warming and filtering the air. Studies have shown that some asthma attacks are precipitated by breathing air that is too cold or too dry for the lungs. Athletes, for example, may suffer an asthma attack when jogging on a cool day before their lower airways are sufficiently warm. But even non-athletes will suffer if their noses are stuffed up -- not an unlikely event during cold season, or for allergy sufferers cooped up in school or at home with stale, dusty, hot, dry air who then emerge into freezing or near-freezing winter air.