Check log files for file transfer
You can manually use the Terminal command cp (copy) or mv (move) with the -v (for verbose) qualifier to keep a log of what is being copied. Verbose only shows file names, no dates.

You should look at man cp and man mv for more details.

Ex:



If you're looking for capturing log data for Finder drags, or disk activity from within apps, I'm not sure how that would be done.