Most states, give
joint custody, unless there is some proof of abuse etc. The amount can vary, as 50/50 to 80/20 etc.
But, if he can prove he lived with her and the kids, and he is indeed the father, I would think he can prevent a move.
We had similar situation, and had a written clause in our
parenting plan that if
BM moved father got custody (also unwed father). (Our case the BM moved regularly, and was unstable).
Sounds to me he can also use her "night job" as a good angle for more parenting time, if he is available during those night hours while she is at work, or times she is sleeping, and children are unsupervised. He should expect to have this time to parent during those hours, as it is better to be with a parent then running free etc. etc.
Our case the BM tried over and over to get sole, and move away, she never was able to do either, as the court always ruled BOTH BIOLOGICAL parents have a right to be involved in the childs life. (Unless abusive situation)
Just make a rock-solid parenting plan, state times for parenting time, rights of refusals, and moveaway stipulations.
Case that states BOTH parents have a right to be equally involved in the child(ren) life.
Davis v. Flickinger (1997), 77 Ohio St. 3d 415, 419. Cordon v. Gordon (October 19, 1987), Athens App. No. 1334