If you keep your tractor indoors, as I do, eventually you figure out putting a dirty tractor away in a clean garage, barn or shop only brings more dirt in. Then you have to clean it up anyway. So, it's kind of "Clean the tractor now, or the shop later... take your pick." It's just easier to keep the tractor clean.
Yep, my tractors stayed inside in my 40' x 60' shop, and I had an air hose right by the door. So if it was just dust, the air hose took care of it. If it was dirtier than that, I used the water hose. I used to have a gun to which I hooked both a garden hose (water) and the air hose, to make a pressure washer type arrangement.
I'm looking to buy an air compressor in the near future not just for the dozens of tires around here but also to clean the dust off and out my tractor.
Do you need some kind of special end on the compressor's hose to do that? My current little portable compressor simply has a Schrader end on it.
I have a short little blow gun like
this with the safety tip instead of the rubber tip. Then I also have a
2' blow gun with the rubber tip. The 2' one was what I used on the tractor. I also have a 4' blow gun without the rubber tip. I don't have a gasoline or electric "leaf blower" or whatever you call them, but I have enough compressor and air hose to serve the purpose.

Of course, I'd recommend quick couplers for all your compressed air tools and hoses. Other than the 3 blow guns, I have two 60' hoses and one 50' hose, a half-inch impact, 3/8" air ratchet, angle die grinder, one quart paint gun, and pneumatic brad/staple gun. Before I moved back to town, I had a few more pneumatic tools, like the same blast cabinet, air drill, etc.