Sooo
For that quartering, running away shot, it would probably not be your first choice? TIC
Most any North American animal, even brown bear can, and has, been killed with the lowly 22LR. Under perfect conditions, with a perfect shot, it is very deadly.
What is a reasonable cartridge to use, is one that will get the job done without perfect conditions or perfect shot placement. Granted, even a 50 BMG will not drop a coyote if you shoot it in a hip.
What you shoot should be able to stop your target even if you are a few inches off, or if the target doesn't present just the right angle.
If you want to use lower powered cartridges, for the game you are after, that is fine, as long as you will not be attempting shots where you are not sure of your aim, or the game does not present a good target at the right angle.