What do I need to do when I change my iPhone 13 to iPhone 8?

Apple’s process for doing this is quite mature and reliable.

It generally depends on your old phone being fully backed-up beforehand …

How to back up your iPhone, iPad, and iPod touch - Apple Support

… and then transferring it all to your new phone.

Make sure that you KNOW your AppleID and password beforehand AND carefully read all the prompts on your new phone. (i.e. make sure that you’re entering EXACTLY what is asked for by the prompt)

Transfer data from your previous iOS or iPadOS device to your new iPhone or iPad - Apple Support

Doh!

Just re-read your post !!!

Don’t think there is any “automated” method available to migrate from “new” to “old.”

While you can easily migrate the phone number by shifting the SIM, moving your data is another matter.

Your 13 saves it’s data in a “newer” iOS than your 8 is even capable of running.

Any transition will need to be mostly “manually” accomplished.