Well, it seem that you are quite curious about the phenomenon. So, as far as I can see, when we do some physical exercise, especially intensive sports, our eyes would water, or it is just because you confronted the direction that the wind was blowing. So, if your eyes stop watering when you stop having sports, then your eyes are good to go. However, if the problem remains unchanged and constant, you should consult a doctor and find our the causes.
It is not normal for the eyes to water when you run. There must be some eye problems for you, like the inflammation which will be caused by the bad environment or the long use of eyes in front of the computers. You should go to the hospital to have a full check on the eyes.
You may get the watery eyes,which happens when the tear duct produces too much lubrication of the eye. Tearing helps our eyes washing away foreign particles that could be harmful.
And there are plenty of reasons will contribute to watery eyes.
It may be a symptom from some type of allergies. Sometimes it can also be because of a sinus infection, headaches, irritations, or even problems with eye-wear such as contacts. If you suffer from any kind of seasonal, pet, or indoor allergies though, most of the time your symptoms will begin with watery eyes. Tearing can happen if your eye is producing too much or too little lubrication.
My suggestion for you is doing a check up ,then you can see what's really going on.