Hi @Ed Turnbull,
Thank you for raising question to Microsoft Q&A.
According to your question, as far as I know - you might need to manually move your MFA from your old phone into new phone, which means you will need to re-register the MFA again on your new phone.
The Backup-Restore function on Microsoft Authenticator will not be able to move the actual MFA from your old phone to new phone. I had this issue once a while ago, so I decided not to involve with that Backup-Restore function anymore. That function was able to only recover the list of accounts in your old phone - for example, you had 5 accounts on your old phone, and if you use this function, on the new phone it will exactly restore 5 accounts. However, the actual MFA did not work for me - when I signed into those accounts, the Microsoft Authenticator did not prompt for anything at all, and I had to contact the IT department/Global Administrator to reset MFA, which allows me to re-register MFA by following this link: https://learn.microsoft.com/en-us/entra/identity/authentication/howto-mfa-userdevicesettings#manage-user-authentication-options
If this is the case, then from my perspective view - with your 3rd party MFAs, it will not do the same thing as I mentioned above. Fortunately, I also had other verification methods to make me not locked out from those accounts. I decided not to rely on Backup-Restore function anymore, or at least I only use it if I have so many accounts that I could not remember at all - and at the end I would need to re-register all of them again.
I hope this helps you with the information. At least you will need to prepare everything beforehand so that you will not be locked out from those accounts when doing the transition.
If the answer is helpful, please click "Accept Answer" and kindly upvote it. If you have extra questions about this answer, please click "Comment".
Note: Please follow the steps in our documentation to enable e-mail notifications if you want to receive the related email notification for this thread.