I don't think you need to "transfer" the file. Using my GF as an example, she moved to a new court district within NS and didn't have to request a file transfer. She recently filed a court application in the new court district and they simply pulled her file from the other area. There was no action required from my GF.
If you moved to another province, that would be a different story. Due to the fact that your file has always been registered with ON, there shouldn't be any problems.