When you create a LiveUSB install, the /boot/grub/menu.lst file contains a reference to the partition on which it is installed, in the form of a UUID.
That UUID is specific to that partition.
When you copy all the files to a different stick/partition, you must change the UUID used in the boot stanzas to reflect that change.
Please note that the changes directory has root permissions, and so would need proper rights to copy it to a different location.
Having completed the copying of the files you can easily set up Grub on the new stick, from any running OS, by using Grub from the command line. How to do this with three simple commands has been posted previously, but if you need it post back.