There is an easy fix ... before moving ...
download the backup .sql file to be used for importing on the new system and edit with an text editor: search for http://oldsite/[folder]/ replace with http://newsite/[folder]/
If you have command line on either server can edit the sql dump there using editors such as nano.
or after moving, use the admin tool for search and replace in Moodle:
http://newsite/admin/tool/replace/
use the full URL to search ... ie, http://oldsite/[folder]/ and replace with (as above) http://newsite/[folder]/
'spirit of sharing', Ken