I got a new passport in Shanghai last year while I still had my RP in the old one. I just had to carry both passports with me when I went to HK to get the new work visa. They punched four holes in the cover so it was obviously officially invalid.

So I dont think you need to do anything about a new RP, its still valid even though its stuck in the old passport. No one dealing with me having two passports even raised an eyebrow about it. 

Bike gypsy is right. That is exactly how it worked for me too. You just have to remember to carry both for a little while.

My passport with my residence permit got ripped apart in a third country. I applied for a new passport and went to China with one good passport without visa, and one ruined passport with my visa/residence permit. I must say, the Chinese immigration control has never gone as smoothly as it did that time.

I had one month left on my RP so I applied for a new one in my new passport. Just remember to provide the entry exit guys with both passports.

...and if your new passport has a new passport number you need a stamped document from some relevant authority to say you are indeed the same person as the guy in the old passport.