Hi Catherine,
I wonder if not preserving Updates "under" a Task when a Task is moved is As Designed (for some reason, such as "when the updates were made, contextually, the Task was part of the former Project, so the updates belong with that Project"), or is actually a bug. Either way, I'd suggest you raise a help desk ticket to find out, and am interested to hear what you learn.
That aside...
If you can create a NOTES report that retrieves the updates in question before you move a Task, if you export them to Excel, then after you move the Task, you could also then adjust the Excel spreadsheet and either kickstart or use our Excel Updater solution to reinstate those NOTES under the moved Task, for context...noting (having done so) that because the Entry Date and Entered By are immutable (e.g. "today" and "you", respectively), that it is helpful to add a suffix to the Note Text with the original values (e.g. "[On yyyy-mm-dd hh:nn by username]...").
Regards,
Doug