Will my currently displaced crown eventually move back up to my gum line after Braces treatment is completed?

The short
answer to this is no. Most likely this crown will have to be replaced once your
braces are finished, once the teeth are moved and teeth are positioned with
good retention. The idea is that they won’t move. If there has been any
movement between the gum line and the tooth crown, this most likely will not
change. So the most likely scenario is that you will have to redo the crown.

It is likely that you will need to redo the crown once your finished with braces to eliminate the gap between the crown and the gums. Braces move the entire tooth and root into alignment. Once the teeth are aligned the gums will not grow down to cover the space and you do not want the tooth to move back up since the point of the braces are to get all the teeth aligned properly. If this leaves a cosmetic problem with the existing crown then the best solution would be to redo the crown.