Hey @Ellenm,
No. You would want to copy not move.
You also want to only copy items that are new, or so I think.
Are you keeping an entire copy of the folder local on your system?
How are you indexing them in DEVONthink 3?
Are you indexing them in situ, or are you importing them into DT's database?
I haven't used @mrpasini's macro, but rsync is a mature and very well tested command line backup and mirroring tool.
I have several utilities for backing up and/or mirroring files, so I would probably use one of them instead of a home grown solution.
Although I might use Keyboard Maestro to launch the process periodically.
What I would do depends upon the urgency of the task and how bombproof it needed to be.
-Chris