I've used SMS Backup & Restore for several years to do exactly what you describe.
Everything backs up to a local folder on my phone and I sync to my home server. The app can sync backups to a cloud storage, but it isn't required.
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.
That looks promising, much appreciated. I should also mention I also mean to back up MMS along with SMS.
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.
This can be done no matter what SMS/MMS app you use, but it is slow.
SMS Import / Export (Imports and exports SMS and MMS messages, call logs, and contacts.) https://f-droid.org/packages/com.github.tmo1.sms_ie/
Neo-Backup with special backups enabled can do that and much more. Works well with synchronization tools. Root required.
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