I'm running on almost new 20" Michelins, and had a blowout on left rear. I mounted the spare, which I found to be a new stock 19" Goodyear which came with the vehicle. It was fine to get home on, but I've got to drive through the mountains to pick up family in Denver today, a 3 hour trip each way. I can't get a replacement tire for a couple days. Will driving on a smaller wheel damage the transmission?

 

It causes more wear on the diff. How much, I can't say.
4WD vehicles should always have the same diameter tires on all 4 corners. When you buy tires, a matching spare should be bought at the same time and rotated in with them. A lot of people don't realize that.