To add to what Ceara stated, when my ex wanted to take our daughter (who has dual citizenship of Canadian and American) to Canada he was stopped three times and asked for a notarized letter stating that I gave permission for her to leave the country. If he didn't have it he would not have been able to leave the country with her. And that is JUST Canada -- by that I mean it is not usually that difficult to get into Canada compared to other countries.
Parents should remember 3 things: Love your kids more than you hate your ex; when you have children the relationship with the other parent is until death; your children determine what type of nursing home you end up in.
Nothing stated by me should be taken as giving you legal advice or forming an attorney/client relationship.
Attorney-GAL in Ohio.
I've removed the knife from my back, polished it, and will one day return it -- long after you think I have forgotten.