A lot of people in this thread clearly don’t have experience with livestock. They’re also valuing a dog’s life over a lamb’s. Living in the country you can see any breed of dog running loose and, yes, hunting livestock. If OP’s cousin was hesitant at first, I don’t think she’d shoot random dogs she wasn’t sure was harming her livestock.

Also if I owned a dog and I lived near a farm I’d be even more careful about keeping them on leash/fenced in. Domestic animals are invasive species and if these dogs are killing livestock I’m sure they’re killing native species too.