No, you won't need to pay another amount of money

If the entire treatment costs $6,000, and during the 1st year the cost is $4,000, you will only need to pay $2,000 to the new orthodontist. You might need to pay a little bit more for transferring, but not a whole lot more. Again, doctors and insurance companies have plans to work around those situations. You are not going to be the 1st family to go through this :)