This topic is close to what is on my mind too, Val. I'd like my children to feel connected to other people, so does moving, or changing schools or grades hurt that? I can see how a long shared experience could be a very nice thing to have. Yet, if someone is not fitting in, not feeling like they have peers, maybe a change will move them into a situation where they can connect with people better. (Maybe? I hope?)
Sadly, I have seen examples where HG+ people did stay with the same group all through school but grew up feeling alienated from everyone. That's something I'll be trying like crazy to avoid with my kids. I'm sure what works best is very personality and situation dependent though.
Here's my own experience: I stayed at the same school K-5. Felt like an alien. We moved to another state where I started 6th grade. Again, I was an alien. Skipped 7th, switched to a GT school for 8-10. Hallelujah! I found my people
We moved to yet a third state. Public high school for 11-12. Yep, I was an alien again. But it was nice to know my people were out there somewhere! I met up with them again in grad school.
I guess my point is that if you're not fitting in, it's worth the risk to try something new. Don't stick to the same group just for the sake of shared history.
