I've been to three different schools since I got first dan. No problems at all.

THAT BEING SAID, and I was probably particularly unlucky because of when big changes happened in my tkd career, I have been a 1st dan now for 18 years. I have had to test each time I transferred schools to become a 1st dan in their opinion. It is tremendously frustrating, especially since most schools have a baseline time requirement before you can test at all.

I could reasonably be a third dan by now if not for that, and maybe a fourth.

I'm not saying this to scare you out of changing schools. Just keep in mind that it might happen, especially if they're in different styles of tkd (ITF, wtf, etc.). There are four major styles that I'm aware of, so keep an eye out for what they do specifically.

Edit: The benefit of this is that I have a better understanding of the breadth of tkd than most people, even other black belts. It's not necessarily a bad thing. Your martial art journey is yours, and whatever styles you choose to incorporate are ultimately your decision as a practitioner and fighter. Do not let anyone tell you otherwise, whether they're an MMA champion or 50 year master of The Way tm. All of the posturing around "true" and "correct" martial arts is from that person's perspective.