Get Your Dog Care Questions Answered by Experts
A couple things come to mind. Your dog can have a blocked tear duct...which means instead of the eye draining normally thru a duct into the nose...it runs down the face. If the dog is active or there are irritants in the air, your eyes secret more fluid and that is probably why you see it after running.
To have it diagnosed...a vet puts a light stain in the eye and then checks to see if any stain comes out the corresponding nostril. Sometimes it is correctable, sometimes people just deal with it.
That is the most probable thing.
Let me know if you would like to discuss if further.