In my case, it was the county and state that the divorce/ court order took place in (Ohio). She moved to other states frequently. I moved to KY for a while.
My attorney told me the case could be moved to the state she moved to, not mine. He also told me I could delay her moves, not stop them as she could always claim economic necessity and the courts would allow it.
So, it'll stay in the original court or follow her around. The court doesn't care about you, so don't get excited about geting custody. The court doesn't care about you getting visitation or building a relationship with your kids, either.
The courts only care that the child support is being paid so they can brag about their compliance rate and the amount they collect every year. Father's rights are not considered.
Also, in my case, money not sent through the courts is considered a gift, not child support. It varies by state but voluntary payments without an order might not count for anything if she decided to get an official court order.
Of course, that was just my experience, YMMV.