DD16 is graduating from a big public school this month (~1600 students). She transitioned from a small private K-5 school (~50 students) following 5th grade. As per usual, the answer to your question is "it depends."
In our case the large public school was open to acceleration options and had been accomodating for other gifted kids already in the system. Also, the small private school - knowing most of their kids were going into the public school after 5th grade - tried hard to instill confidence, independence, and the ability to self-advocate into the students.
In our case (and with the benefit of hindsight/confirmation bias), the transition from small private to large public worked out great. Sure, it wasn't all smooth sailing, but all-in-all, we're glad we switched to the larger school system where there were more kids like our DD and where there were ample opportunities to try out new academic interests.
Research the new district(s) - try to find one that will work with you and your children (these districts do exist!). In transitioning from private to public, we met with couple of different districts and open enrolled to a neighboring district because if just felt like the right place for her.
Best of luck,
--S.F.
