android-conversation-backup
Backup and archive text, pictures and other multimedia messages and call logs.
This tool can backup your SMS messages, MMS messages with included attachments, and call records, into a zip archive. You can specify a set of numbers to limit the conversations backed up. Optionally, you can request that those archived messages and/or call logs from your device are deleted.
Once backup is done, you can share the zip archive, allowing
you store it using any apps on your device which can share (e.g. email, gmail,
Google Drive, DropBox, Box, etc).
Finally, you can review the prior backups done, and delete them or share them
again (re-send them somewhere).
Primer / Documentation: http://ugorji.net/project/android-conversation-backup
Blog Post: http://ugorji.net/blog/android-conversation-backup
Download on Google Play: https://play.google.com/store/apps/details?id=net.ugorji.android.conversationbackup
Development Notes
Google doesn't provide public API's for accessing SMS and MMS.
I had to reverse-engineer the actual android src to know what to do. However, this has been tested and works on Androind 2.3 up to Android 9.0.