They mostly don't catch diseases from other quadrupeds, but they could certainly carry germs on their feet, tack, etc.
Oh, llm just posted along similar lines.
when horses travel they are usually going to shows etc where they are in contact with other horses but unlikely to be in contact with other farm livestock.
Can't agree with that bit, llm. Horses go off trekking all over the place, hunting too, and many shows are held on farmland anyway!
Thankfully I think most horse riders are sensible and sensitive enough to not ride bridleways across farmland during a disease outbreak, but they could legally do so if they wished.