The children stay where they are when the papers are filed, until the court rules, although it's a good idea to arrange for visitation for the other spouse, if the court date is more than a week away.  Children need a good relationship with both parents, even during and after a divorce (if not more so), for their best emotional health.

Child-snatching, and denying visitation without a very, very good reason, are bad for the children, and give the court a reason to take custody away, or deny it, to the parents who do that.