how to retain Time Machine history of a file when it is moved to a different volume

Background to question: a disk image (dmg) on my macbook’s hard drive  became mysteriously full (with miscellaneous used space according to GrandPerspective) and I could no longer save to it.   I deleted a couple of large files but Finder’s info is still showing zero bytes available.  I'd be very interested to know the best way to deal with this. So far I have simply used Finder to copy the content  (23 folders and 2 files)  to a fresh directory on the hard drive.  The dmg file system is Mac OS Extended (Journaled); the copied content is 733.8 MB on disk for 3,596 items according to Finder.   According to Disk Utility the dmg’s  Free space is 442,609,664;  its Used space is 3,213,438,976; and the file count is 5,845.    (GrandPerspective scanned  3765 files.)

Any pointers or advice about this general problem would be appreciated but my immediate question is how to retain Time Machine history of a file when it is copied to a different volume.