If you can live with more Internet traffic and less privacy compared to what Duplicati provides, letting Google Drive take care of sync between computers, then backing up one of the computers’ sync areas gets close to your wish. Omitting the flash drive means you might have to look to see if sync upload is done before leaving a computer, and similarly at the computer you arrive at, there would be some initial delays to get file changes downloaded. I’ve also seen some oddities where a file conflict is not “resolved” the way I would like, but that’s possibly a result of mixing computers – possibly your use which seems largely sequential would operate well.
This avoids keeping three independent backups which are largely the same except for latest file changes. At least with Google Drive, recent versions of files can be obtained through the drive.google.com web interface. Note that this is rather cumbersome if you have many (or all) files to restore. In that regard, Duplicati is nicer. Ordinarily I’d worry about ransomware, but the chances of it hitting all computers at the same time seem slim.
EDIT: However automatic sync can propagate file damage into the replicas, especially if they’re powered on.
If you like Duplicati’s security and privacy, and are willing to do scripting, Is it possible to set a date limit? may provide a way to backup recent changes, while letting the flash drive handle the less frequent computer sync. You would also backup a computer with Duplicati daily to supplement your new-file-only backups with old files.
There are many ways to put this together. If you look into other backup and sync programs, it gets worse. 