By statute, when a child is born to unmarried parents, the mother has sole legal and sole physical custody until there is a court proceeding determining otherwise. (In comparison, parents have equal rights to the children when they are married until there is a court proceeding determining otherwise). The father should probably be talking to lawyers about bringing a custody action to establish his rights.
Depending on when the mother and children left the state, the custody action may need to take place in the new state. The action will be decided in the child's home state (as determined by the Uniform Child Custody Jurisdiction and Enforcement Act). If they just left, Minnesota may still have jurisdiction, if it has been 6 months or more, the action will probably need to be brought in the new state.
Father would need to start a paternity action to have a court declare him the father of the children. Until he is legally declared the father, he has absolutely no legal rights to the children. Through a paternity action he could ask the court to rule on the issue of custody, child support, parenting time (visitation), and whether or not she could move the children out of Minnesota. If there is a court order for custody, MN does not allow a parent to move the child out of the state without either the other parent's consent or a court order in which she would need to show the move is in the best interest of the children and it is not being done to interfere with dad's parenting time.
He really needs to start a paternity action in order to have any legal rights. This is true even if he is ordered to pay child support as you can be ordered to pay child support without the court making any custody determinations.