My horse had something that sounds similar last year- it started out with a few globs of nasal discharge and pretty quickly turned into coughing any time I rode him faster than a walk. We tried watering his hay to cut down on dust and added AniHist to his feed at first and it continued to get worse. I switched to Cough Free and it went away within a few days (I also tried taking him off it a couple times and it came back, so I knew the CF was doing the trick). I did have my vet monitoring him to make sure it wasn't an infection (and didn't develop into one).
I was able to stop the supplement sometime in October without him going back to coughing (good thing since he started refusing to eat it!) We thought it was seasonal allergies, but it didn't come back this year.