You might look for another orthodontist to continue your treatment because you are moving (to another city, for example), or for similar reasons on the orthodontist's side (illness, a long unavoidable absence from work and so on). In that situation, the orthodontist undertakes to suggest a replacement who will take over your case. Of course, you can also seek another opinion on your treatment if you have lost confidence in your current orthodontist. That is your right, and you can change orthodontist during treatment. You may be told that the appliance currently in your mouth will need to be replaced as a condition of continuing treatment. This is because there are many types of braces on the market and each orthodontist chooses which ones to work with. To continue your treatment with the braces already in place, we need to use the same type in our own practice, so that we can rebond a bracket if necessary and have the right set of archwires for that type of braces. The difficulty in finding an orthodontist to continue your treatment comes from the fact that the specialist you choose probably doesn't use the same braces, and it is not financially worthwhile for you to have a whole new set of a different type fitted.
With Invisalign aligners, changing orthodontist is not a problem – you simply need to choose another provider of the same type of aligners.
This answer is for general information. Your own case needs an examination by an orthodontist.