While I have never put down a donkey I have put down several horses. It's the same as hogs and cattle. A 22LR works fine if you hit them in the thin spot going into the brain. I've been told from the side in the ear works the same but I have never seen or done it. I started being the designated shooter for hogs and cattle at 7 or 8 years old when my uncle was having a hard time seeing the front sight. The extended family would get together and slaughter hogs and cattle in the spring and in the fall. When I was 7 or 8 he missed the mark several times with the 22 LR rifle on a steer and the bullets did flatten and basically fell straight down. That was a very long time ago and I don't remember if there was any bounce but I clearly remember watching the bullets flatten and fall. I have used a 22 Mag a couple of times on baloney bulls. I did use a 7mm Remington Mag on a car struck bull once. I did use a 12 gauge slug on a couple of others hit by cars. As a kid a worked 2 1/2 as a very small-town gas station. Car cattle, deer and antelope collisions were common. The wrecker had a single shot 22 and a single shot 12 gauge for that purpose. While technically not legal for game animals game warden and cops had no issue with putting down animal at the scene of a collision.
Again I have not seen it, but I have heard same failed results the with 38/357's when the hits were off the mark.
I have watched Buffalo being slaughtered with a 375 Winchester Levergun. Missing the brain will penetrate but it will not put them down. That resulted in a charge and the second shot hit the mark and did the job. Dad was the shooter. I still have both bullets.
With the low cost of subcompact 380's that is the smallest SD round I am willing to carry. I do carry a 22 LR as a camp gun at times but the intend targets would be rapid skunks or coyotes. If its two legged SD its 380 or larger.
Good video here on the 25 ACP
https://americanshootingjournal.com/...%20its%20track.