I've been looking around to find a good, privacy respecting way to sync my messages between phones. I decided I'm going to use SyncThing so I don't have to mess around with a server. The only problem with this is that I haven't been able to find any apps that work on modern Android that routinely backup and import messages from a file/folder into the messages database. Does anyone know any app that might do this?
Same. This is exactly what I do. I'm on my 5th Android phone since 2009, and I still have all my call logs and SMSs going back until then, because I backup from my old phone and restore to my new phone on every upgrade.
While it isn't routine, I quite like QUIK, as it allows at least manual backup of messages to a folder you set within your phone, it can be found on F-Droid, although I'm unsure if it's within any repositories or just found within. QUIK
My apologies it isn't exactly what you're looking for.
Then I'm glad to be of at least partial assistance with your search. Ah, I'm not entirely sure if it does, and I can't presume so due to it including the entirety of the messages, and I unfortunately can't check, as I don't have any MMS to save.
I do however hope it does, but if not, I wish you the best of luck with your search, and that others will chime in with more effective solutions.