I've seen three different versions:
1. New student comes in, keeps rank. This was many years ago when I did TKD/Kickboxing. New student came in, Shotokan, IIRC, and she was allowed to keep her rank. She was even allowed to stand with the upperbelts. That didn't go over too well with the rest of us, she had a bit of an attitude.
2. New student comes in, gives up black belt, trains with the rookies.
If the styles are totally disparate, like going from say TKD to BJJ, it's not a big deal. If it's going from say TKD to TSD or Karate, I think it's a bit unfair. One of the reasons there are beginner and advanced classes is to separate levels of ability. It creates a sense of security for new students. In the case where the styles are similar, I like the next option:
3. New student comes in, gives up black belt, trains with the upperbelts.
In similar styles (IE: TKD and TSD), this recognizes that the student has a certain level of skill, and may only have to learn forms or certain techniques and doesn't have to reinvent the wheel on kicks and punches, for example. It's win-win, I think. The new student doesn't have to feel as though every last piece of former training was useless, it recognizes that this student can handle themselves, and recognizes that this student is not a beginner. But the student still has to earn the right to wear the black belt of the new system.