Our experience was that every time we moved, the old orthodontist would give us a refund for the time remaining in treatment. Then the new orthodontist would charge us what he felt would be the cost for the remaining time. We moved five times when my son was in orthodontic care, twice after his braces were on. Both times, the new dentist charged significantly more than what we were refunded by the previous orthodontist. It was VERY frustrating.
One key to making it less painful, is to get recommendations from the orthodontist you are leaving for those in the area where you are moving who use the same technique and appliances. That helped us so much with our 2nd and 3rd move. The 4th and 5th were the nightmare. It didn’t help that the orthodontist we were referred to with our 5th move refused to see my son because he was in the middle of treatment and instead forced us to see his partner (who, it turned out, was NOT the same quality as the older gentleman). So much for supporting our troops!
Our experience financially and with care at that last orthodontist led us to choosing a completely different practice for our daughters. (My husband is retired now, so no more moving.

)