Does a pygmy owl migrate in the winter?

No it does not. it's a resident bird. They survive quite well during the winter and have no need to migrate, much to the dismay of city dwellers where these birds live.
Pigeons, unlike some other species of birds, do not migrate. They can and will fly many miles away from their roost to find food, but at the end of the day they will always return home. When pigeons are a few months old, they imprint their location in their brain as "home" and will always return there.