Coming from an owner of two lurchers as well as an English setter and two cockers, let's get real here. Pet dogs do flush birds, and chase them, and pet dogs also catch wild rabbits (if they are fast enough !!).... and if they kill it, fine, that's nature ( know I don't like it either but hey that's life) - if they don't then you either release it if it's ok, or kill it yourself if its injured. All part and parcel of being a dog owner. Spose it depends where you walk as to what they might catch - a park in London might have squirrels and pigeons, where I walk is rabbits, pheasants (Barley flushed a beaut today !) and the occasional deer.
Totally different thread is needed if you are discussing what to do if your working gundog does the same though.