Most of my work involves projects being moved from edit to online for me to grade etc. I have not had this problem when simply moving the project via the original external drives. If you need to move to an alternative external drive, I recommend using Media Manager to make a copy of the project to the new drive.
This avoids having to relink offline files. I have seen that error message when people just copy folders over. Not sure why but relinking is such an unnecessary pain and too often files are missed – usually a bit of music or a graphic that was saved to the desktop.