Under what circumstances can a man who was never married to a child's mother prevent her and the child from moving out of the state?

the best defense is to have some physical documentation stating that you are indeed the father. once you have this get a lawyer to argue before a judge that this move would not be in the best interest of the child and due to physical constraints you would not be able to be a stable presence in the child's life which you strongly desire. this is after you are proven to be the father and have a custody agreement.