Exercise can definitely contribute to asthma as well as seasonal allergies. I have suffered from both the past several years. If you have any form of asthma you need to consult your physician. You will most likely be prescribed a "rescue" inhaler. Keep this with you when you run. You probably wont notice any allergy/exercise induced asthma until you finish your run when you are cooling down. If you have trouble breathing during your exercise you need to stop immediately until you have consulted with a physician.
Like I stated before, I have allergy induced asthma. It usually hits me during the spring time and tapers off when summer arrives. It has never hampered my running. Recently a friend suggested I use a nasal saline rinse to flush out my nasal cavity. At first I thought he was crazy. I used it pretty regularly during the first several weeks this spring. Since using the saline rinse I have not needed my inhaler at all.
Once your lungs become accustomed to running/breathing your symptoms should diminish significantly. Also you should use caution while exercising in the winter months. The cold air can also induce asthma.