I am not sure that I understand what you are asking. Custody and child support are two separate issues. One does not affect the other. Any change in custody would require a showing of endangerment, either physical, developmental or psychological and a demonstration that the benefit of the change in custody outweighs any harm caused by the change in custody.
I cannot understand why any parent would not want to share custody, legal or physical.
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If the main goal is to be able to move out of state, that can be accomplished without having to change custody. Most likely the parenting time order would have to be revised though. However, these potential changes are not dependent on whether or not the other parent is paying child support.
So, you could start by seeing if the other parent is willing to revise the parenting time order and allow the relocation out of the state. If there is no agreement, you will have to prove to the court that there has been a substantial change in the circumstances and that a change of custody and/or parenting time order is in the best interest of the children. It may be wise to consult with an attorney to go over the facts of your case in more depth.
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The post is not clear and unambiguous. If you are facing these issues I urge you to seek private attorney counsel at this time. You can discuss your specific goals and how they might be attained. We avvo attorneys are available. All the best. Tricia Dwyer Esq.
MINNEAPOLIS/ST. PAUL and outlying areas to ST CLOUD. Do seek legal counsel for your personal legal issues and needs. This post is not legal advice and does not create an attorney-client relationship. This post is to be considered general information which may or may not apply to your personal situation.