Both parents have joint custody. Mother and father never married each other. Mother moves out . Child stays and lives with father it considered abandonment on the Mother? Also mother moves out of state Child is living with his father. Father gets Married. Who has rights over child

Abandonment is generally a criminal charge for leaving a child with no adult supervision. So that's not the case here. In a few states failure to support a child may also be called abandonment. But you don't indicate where this is (ANY question on law needs to include your general locale as laws vary by area.) or whether there is court ordered support.

You say both have joint custody. But you don't say if its legal or physical or both, or whether its court ordered.

The mother moving doesn't affect anything as far as I can tell with the limited info you gave us. Neither does the father marrying. Both bio parents still have joint custody.

If you tell us what either the father or the mother wants out of this situation, we can advise further. Abandonment may be used as grounds to get something else. But whether abandonment exists wouldn't seem to matter on its own.