if it involves joint custody
Yes. However, if the visitation is no longer practical, one of the parents may need to file for a modification of custody or visitation.
If the other parent have visitation rights you will need their consent as well as the courts.
Leaves them permanently or in violation of court ordered custody/visitation time? In either circumstances, the non-custodial parent may file an action for contempt of court against the custodial parent and/or file for custody/visitation modification based on the same.
The court orders must be followed. If the parties have come to a mutual agreement they should file it as a modification to the visitation and custody orders.
If a custody order is in place outlining visitation, no, not without being in contempt of court. If no such order is in place, either parent has the right to file for one or modification to an existing order at any time.
Both parents cannot be awarded full custody at the same time, however they may be awarded joint custody. If that is the case, then the parent in Arizona needs to file for a custody/visitation modification based on the father's move, if he doesn't intend to abide by the original agreement. If he does, it is up to him to pay all related expenses in coming to visit his child or taking them to Texas for his parenting time. Also, Arizona law provides that the parent intending to move must file a notice of intent to relocate prior to moving and such a move be approved by the court. If he did not do this, he may have his custodial/visitation rights modified based on the same and your child support increased as a result (you would have to file for modification of this separately).
In the US the law says that lesbian partners have the same rights to custody or visitation as their heterosexual counterparts.
The mother has the option to file for modification of the original custody/visitation order if the parent moves or even prevent such a move if state law has that provision, however the other parent has the right to respond to such a motion. Ultimately, the courts want to maintain involvement of both parents in a child's life. If the parent moves and maintains visitation as outlined in the original order on his own dime, there would be no reason to give the mother full custody. However, if the move would make it impossible for the parent to maintain visitation as originally outlined, a modification may be granted.
You cannot force a parent to visit their child, however if they are not abiding by the court order outlining visitation, you can either file an action against them in the court of jurisdiction for contempt of court, or apply for a modification of the original custody/visitation order, stripping them of their rights to visitation or lessening them. If such an order is approved and the custodial arrangement is changed, child support may also be modified based on the same.
If you are not married and there is no custody or visitation order, she has custody automatically. The father have to prove paternity in court by a DNA test and then petition for custody or visitation. He can then also pay child support.
You may arrange visitation through the courts.
Check Link BelowConsidered Factors in deciding to go for a child custody modification?