My granddaughter is being moved from MN to Montana because her mother found another new boyfriend, mind you this is the 6th one in less then a year. There is a court order for support and visitation in effect but the parents have never been married. Is it legal for the mother to just up and move?

If there is a court order for visitation and the move would make it difficult for the father to exercise that visitation then the mother needs to get permission from the court to move. And the court will be unlikely to grant it if the father fights it.

If the father has evidence that this move is planned he can go to court to get an injunction to prevent it.

If the child's father agree's or if the court order for visits allows it. Does the current court order say anything about her moving,

If not, the child's father ( esp if he has actually been visiting the child) can file to stop her from moving with the child.