There have been several times where my horse has caught a cough due to another horse having it. The worst was when she looked rather sick,and when I asked one of the women who worked there, she said "She just has the sniffles, it's nothing to worry about." I was a very worried mom. I made sure she got plenty of liquids and was able to walk just a bit. I also cleaned her nose quite a few times too. We didn't get a vet out for that, but the people who worked there said they would give her something. Not sure if they really did.
If you are concerned, you could always call up a vet and let them know what is going on. They would either say it's nothing to worry about, or they might say they'd like to check it out.
How soon after you moved your horse, did he get the cough? It might be that the move lowered his immune system and that's how he got the cough. The other horses might not get it if there immune system is good.
But I would call a vet and ask their opinion on it.