I had a 96 Subaru Outback for a while. Got a flat on my way home from work at the right rear corner. Pulled off the road, changed the flat out for the doughnut spare, and drove 11 miles to a tire shop. Had the tire replaced, and got back on the road. By the time I got home, I had a constant grinding noise coming from the "center Diff", and a loud popping coming from the rear axle. If you have an all wheel drive or full time 4 wheel drive vehicle the tires must be the same size. When I took the outback in to have it fixed, the mechanic showed me a fuse holder under the hood. He told me that if I had pulled the fuse in that fuse holder it would have disconnected the center diff and no damage would have been done. However, Subaru did not tell you that in the owners manual for 1996 models. I beleive they changed the owners manual at a later date to include that information.
My wife drives a 2010 Outlander GT with S-AWC. She dropped it off before work one day and had two new tires installed on the front wheels. It needed 4 but we only had the scratch for two for whatever reason at the time. She picked it up after work and drove less than one mile. It lit up the all wheel drive caution light on the dash. The tires she had put on were the same tires it came with from the factory, but the old ones on the back had worn enough that the traction control sensed different speeds between the axles and disconnected the all wheel drive. She had to take it back and have them put two more new tires on the rear, and put it on a credit card. (we only use a credit card for emergencies)
I have an FDii Pat. Fullsize spare. Since that episode with the Subaru, each time I have had to use the doughnut spare, I have always made sure I installed it on a non-drive axle. If it was on a front wheel drive car and a front tire went flat, I would jack up the rear put the doughnut on the rear, and then use the good rear tire on the front drive axle. That way anything connected with gears or clutch packs was spinning at the same speed. It was not cheap to replace the center diff clutch pack and rear differential on that outback. It's not an expense I am willing to pay out again. If all I had was a doughnut spare, I would swap it out for a fullsize steel wheel and used tire. It's not hard to find that combo for around 100 bucks, and it could save you thousands, literally.