Back in the 60's on the family farm we had a pack of feral dogs move into an abandoned farmstead near us and they began to come after our sheep, small hogs and chickens. We tracked them to an old storm cellar they had made into a den and went after them with my Grandfather on a 12-gauge Remington pump and me with a Springfield-Stevens 87A loaded with .22 Long Rifles which allowed it to fire semi-automatic. We were not very successful due to the dogs having a guard system to warn of our approach and the limits of our firepower. The shotgun was effective within ten-yards (using field-loads) but did more wounding than killing. The rifle bullets (lead, roundnose) seemed to be deflected off the skulls in full-on head shots and had surprisingly little effect. I managed to take a couple down who were running across my field of fire at 50 to 75 yards range, but it took a salvo to manage the job. These weren't really big dogs (or chihuahuas either!), perhaps averaging a little larger than most coyotes I've seen and some were heavily furred which might have affected penetration. It would be my opinion, based on this long-ago experience, that regular .22LR would be of marginal value unless you could manage to get very, very close. Fragmentation rounds would not be very effective since penetration to vital organs would be the most efficient at killing the beasts. Head shots would have to be at a ninety-degree angle to the flattest part of the skull since otherwise it would deflect off. Making a kill shot on a fast-moving, fairly small animal was some of the hardest shooting I have ever done and I guarantee that if you don't get an effective shot when the thing is sitting still, your chances for a take-down drop to nil. A .22 WMR would give you a better chance of success. Good luck.
"History teaches that war begins when nations believe that the price of aggression is cheap." - Ronald Reagan