OK, Yort. Don't worry about the process, I see you now.
When you say "shared custody" what do you mean?
How old is the child?
Can you give me some more general details?
What you have is a status quo of shared custody without any court order or domestic contract.
You're right in that the move has to be taken in contexts of what's best for the child. If the mother has no reason to move, then she ought not be taking the child just because she wants to. Having said that, she must have her reasons. If she wants to spend her time with a new boyfriend she met online hanging out in a commune picking magic mushrooms, that's a far cry from having a good job or an opportunity to go to school or something similar.
Neither parent should be unilaterally changing the status quo unless that parent has the specific right to do so by court order or contract. In this case, if father is that involved then father has two options. If the mother has told him outright, or said in writing that she intends to move, then father ought to get a lawyer to write her a letter informing her that she doesn't have his consent to relocate the child, or go to court for an order that the child's primary residence is not to be moved out of the jurisdiction without the consent of both parents or a valid court order. If father has only heard through third parties that she plans on leaving without telling him, then he can still go to court or see a lawyer to raise the issue with mother. Either way, it ought to be communicated to mother in writing as soon as possible that father is not giving his consent.
The fact that regular child support is paid only helps to show the father's commitment to the child, although support and access/mobility in law are separate issues.
Go see a family law lawyer right away. Good luck.
please accept this answer so that I may receive credit for the time and attention I've put into your matter.