We changed schools mid way through 1st grade and it ended up being a great decision.
The deal breaker came when my son received advanced work without instruction- due to some maturity issues, he didn't know how to ask for help. I brought it up with the school, and they responded by pulling his advanced work.
I was also frustrated because his test scores came back very high (school's own test) and they didn't respond until I made some noise (from a school that advertises differentiation right on their website).
I'm no longer upset or even critical, these are just the facts. I expect that schools do underestimate highly gifted kids, because many of the kids on this site are the 99.9%. They are few and far between- it's understandable that instructors would default to the more common explanations for disengagement, disruption, inattentiveness, and all of the things kids may do when they are bored.

Proactively, I began looking more closely at the math and language arts curriculum. It was clear that graduating sixth graders would have ample math and writing skills, but it was nothing special. My DS hated the math curriculum and we wanted something better.
I worried about emotional issues with the change- in our case, it turned out fine, with a few rough patches. DS settled nicely when he was challenged and engaged throughout the day.
I would look closely at the new school- try to understand your child's needs and learning style as much as possible. Mine is self-motivated, intense, creative, and competitive. He was dying on the vine in a traditional classroom of phonics, uninteresting reading, and math facts. Montessori is working for him in these early grades.
We chose his new school because we felt they were nimble enough to respond to his academic needs now and down the road, and to help him manage perfectionism, social interactions, and anxiety.