Ditto to what the other guys said...
To add to it, I've met several players that when you ask them where they used to train they'll say something like, "I started in tennessee and got my blue belt from Joe Jack Garcia, then I moved to Chicago for a few years and got my purple belt under Billy Bob Santos, and now I'm here..."
When you grow up (in the US), you usually go to preschool at one school, then elementary school at another, middle school at another school, high school at another, and college at another. All of these schools are a part of your heritage. Some people are lucky enough to train under the same instructor for their entire BJJ career, but it's pretty rare. When we change schools, it's like getting divorced and re-married. You still have ties to your old family tree, but now you have a new family tree to add to your old heritage.
And yeah, any time you go to a new school, whether you're moving or just doing a drop in, expect to get thoroughly tested by the guys around your same level. We've had several guys from out of town schools come to our gym, and some guys see it as an opportunity to test OUR gym against THEIR gym.
Just be loyal to YOUR schools, whether it's your old school or your new school, and you'll be fine.