I really believe it starts out as a territorial thing. The birds are in their yard - their space. Anyway, you know what happens to anything that gets in a yorkies yard!
My neighbor, a few houses down has a bird dog that got out a few times. Kacee would see him running around a big mugo pine in the yard. He would sit up and try to see if there were birds in the bush. After those episodes, Kacee would go running around that pine tree and sit up just like the bird dog did trying to find birds. She was imitating him.
I have saved a few baby birds from her that got into her fenced yard. Unfortunately, I couldn't save the baby rabbit from losing a chunk of hair and skin.