OK. Thanks for the information. While it is difficult to predict exactly what a judge will do, especially considering I don't have all of the information about your situation, I'll tell you what I think based on my experience.
First off, one of the most important things in a child's life is stability and consistency. Judges know this. As such, when confronted with issues like this, most judges tend to err on the side of stability for the child, all else being equal. So, from what you've said, that seems to weigh in your favor.
Second, judges don't like people who put their own needs before the child's. Here it seems as though your ex is infatuated with someone she basically just met and wants to uproot her children to be with him. That makes her look like a parent who's lost sight of what's important, which, of course, will also weigh in your favor.
So, based on what you've said, it appears that you have a strong case to keep your child with you. If she tries to move off and enroll them in a new school district, you need to haul her back in front of the judge that set the custody rules and address seek custody. At the very least, you need to tell her up front that is what you plan to do, which may cause her to rethink the entire plan.
Again, that is based on the few things you've told me and I'm sure there is a much bigger picture here, but that's my impression.
If you have any questions, feel free to ask. If not, please remember to "rate" my answer. Good luck!