Why should a non-custodial parent pay child support when the other parent moves out of state for no reason and doesn't get visitation?

Because its their child and the child needs to live regardless of the actions of the other parent, second its a court order and those should be followed. If the non custodial parent is unhappy they should go back to court.

Because the court order says to, that's why. The question really is why did you let her take the kids out of state. You can sue for them to be returned to live with you. She can't take them out of state without a court order or your approval. That assumes there is a custody order in place.

You need to sort that out with the Court by filing a motion to terminate child support because the Mother is in violation of the Judgment of Divorce.

Because child support is to help provide for the support of a child the nc parent helped create and bring into the world. It is not a cover charge for visitation time. Most states have laws that govern relocating out of state and those laws are present to help preserve the ability of the nc parent to maintain a connection with the minor child(ren). One must invoke such protections to rely upon them. If one does and the move is permitted by the court, an appeal is possible. If the protections were not invoked in advance of a move, it doesn't mean visitation is lost forever. However, terms may be less favorable once a new residence for the children exists.

The Courts treat child support and child custody/parenting time as two separate and distinct issues. You should ASAP file a Motion with the Court if the other parent moves out of the State without permission, and you should also ASAP file a Motion to enforce your denied parenting time. It is important for you to understand that, just because the other parent is wrongfully denying parenting time, does NOT give you the right to withhold Court-ordered child support.