How to automatically move files from one server to another servers - not syncing but moving

I worked for a company that had this as in internal workflow. We had a Perl script that would use rsync over SSH on the backend. Once the copy was complete (verification of md5 before and after copy), it would delete. Of course, it wrote a log.

You're asking for a product recommendation which is off-topic for this site, but here's a way you could do it yourself. Scripting of this sort is entirely within the realm of duties for a systems administrator.

A "move" is really just a "copy + delete"

Any commercial product that does a "move" is just going to do:

• Copy file FROM source TO dest
• Verify copy on dest
• If verification passes:
  • Delete source
• Else:
  • Exit with error
• Output results of all steps to a log

These steps are all easily scriptable on any modern OS without adding any additional tools.