It should. I've just done some testing to verify. If I create some bogus
text, apply a flag to it, then cut/paste that flagged text into another
location on the page, another page in the same section, and another
section/folder, the text is pasted, and the Notes Flag Summary is kept
updated.
What are the settings on the NFS Task Pane for "Group Notes Flags By" and
"Search:"? If you are only setting the NFS to search the current
page/section, you won't see those moved notes when they're moved to another
section not being monitored by the NFS.