I always let the different style 'carry' their old rank. But testing going forward would have to meet our requirements. It can make for a long time before moving forward in rank.
At shodan I changed teachers within the same organization. The new teacher and his students were so far ahead of what I knew, I felt a little embarrissed with my new rank, and suggested I just put a white belt on again (there were no colored belts). He insisted I keep it - I had earned it. For my Nidan test I tested under his requirements and went forward from there.
This is the way things look to be handled in the CAA division 1
(I don't have experience in the other divisions). Bring your rank but leave everything else behind and start anew.