Dogs can travel far when they escape, and you might not know you have some near you until they attack.
Like kessel said, dogs travel a long ways in a short period of time. Just because your close neighbors don't have big dogs doesn't rule out dogs, and I'd be pretty sure it's dogs, just one or two could create that damage. A coyote would have taken at least one with it when it left to eat. Only predator I know that will kill multiple birds without eating them is a weasel and that's not weasel damage, they will just take the heads off.
I quite often get two different dog visitors to our property, each one lives over a mile away as a crow flies. They and their owners are lucky the dogs have not been interested in our chickens, just like to pee all over and annoy the heck out of me.
I would have a hard time shooting someone's dog, not the dogs fault it's the owner's, but you have every right to.
I would start by warning and demanding payment for damages. Repeat offenses would not be tolerated though, they would be 'eliminated'.
Trail camera would come in handy, catch the perps in the act with concrete picture proof to show the owner's. I wouldn't accept less than $20 a bird for damages. Time, feed, care, your time, replacements, $20 a bird they are getting off cheap.