Short answer: When transferring from one doctor to another there is sometime a record duplication fee, that fee is minimal and charged to you from your current Doctor. The doctor you are transferring to will not charge you to transfer in but Will charge you something to move forward with the treatment.
I tell most of my patients, with braces, to expect treatment time to be lengthen approximately 6mos and add approximately $1000 additional. With Invaliding there is usually just a monthly charge until you complete treatment, then your retainers maybe additional.
One final thought, if you paid you initial doctor in full, sometimes there is a partial refund.