Foreign animal in your yard? Don't you have the right to "restrain" said animal, and call the local authority to retrieve it?
When your neighbours have to start paying $X every time they let their animals stray, they'll soon do something about it.
I had to restrain a neighbour's dog once for hassling my free-range hens. I didn't mistreat it, merely grabbed it by the collar, walked it to the neighbour's place, and advised the neighbour of my rights regarding animals on my property - said rights including shooting her dog if it was hassling, attacking, or even playfully chasing my chickens. She wasn't aware of the rules concerning domestic and farm animals in rural areas, and, to her credit, apologised and promised to never let it happen again. And it didn't.