migrating from iPhone 15 pro to iPhone 13 mini does not work

You can go forwards, but not backwards.

You can update or restore from the backup of an older phone to a newer phone.....but you cannot use the backup of a newer phone to try to update or restore an older phone.

Is the old iPhone iOS updated? Not sure if you’re saying it is attempting to update iOS first and that’s stalling (which may be because of the servers being congested) or if you’re saying the actual migration is stalling. If you can’t wait you may need to either do an iCloud backup or connect to your computer, do a backup that way and then restore that backup on the old iPhone.

So if you’re talking about back up iCloud restores? What I normally do when I go back-and-forth between devices I set up the new phone skip all the set up, upgraded iOS upgrade to the latest Denny erase reset settings then go to the new phone I want to use or vice versa And then of course follow the set up process from there making sure both devices right out the gate are updated with the same iOS you can restore from cloud back up or symmetrically. Use the mirror image as long as both are on Wi-Fi 😊

How exactly are you trying to "migrate"? Quick Start? Never worked for me trying to move data from a newer phone to an older phone.

So what is your solution?

As I said......this "method" never worked for me in the past. Maybe another user who has had better luck can help.

Do you have access to a computer? Have you done an iCloud backup? You will want to backup the new phone to avoid any oops moments if for some reason your data is erased.