Don't know if this is exactly what your asking for, but I had to register my new passport today (4th Sept) and I will need to get a new Carnet issued as well. I do not know if you have a Carnet or not, but if you do, then the number on your new passport will not be the same as the passport number on your older Carnet. If you do not have a carnet, then the procedure may change slightly, but not much, because, except for using my carnet for entry and confirming the old passport number on their information screen, it wasn't required.
The procedure is quite quick and painless if you do this at Migraciones in Breña.
You will need your new passport and a copy, your carnet (if you have one), form F002, (available just inside the front entrance) and a Banco de la Nacion receipt for $/19.40. (nuevo soles)
Go to the information desk first. With the new system this is where you will normally need your carnet or Passport for ID. A person will be in attendance with a personal screen. Entry to The Information Desk is by showing your Carnet/PP and they will issue you a ticket with a number, then just watch the screen. Once called, explain to them your situation and you can ask, or they will decide to give you, an instruction sheet for a "traslado de sellos".
This has three instructions. 1. your new passport and a copy 2. form F002 and 3. Bank Receipt.
The form F002 has a spot about third from the top to mark "traslado" the rest of the information is basic, name, address, etc.
Go to the Banco de la Nacion and show them your instruction sheet and pay. Get a copy of your PP if you haven't done it by now.
Go to the steps where you go up to the third floor and you will see another person with a personal screen where you need to show your Carnet/PP and instruction sheet again to proceed. You will get a ticket with a code and a number. Go to the third floor and they will direct you where to wait. Watch the screen for your number. There is also a person there to help you if needed. Once called give all your information to the person at the desk, do the finger ID and they will tell you to wait again. When the backroom staff have done their bit, your passport will come back and they will call your name. You will have a new stamp on the back page.
Accepting that I always get to Migrations early (7.45 - 8.15 am) because there is hardly anyone there at that time, the whole procedure took me about thirty minutes and about 12 minutes of that was waiting for my passport to come back. This also included doing the bank and writing out the form.
If you do have a Carnet, then getting a new one issued is a different procedure, but at this point, they do not take it from you because your carnet number is now written on the new stamp in your PP. It will need to be replaced at the next time of renewal.
Hope this helps if you haven't already done it by now.