So it sounds like you have your games on something other than your external, then yea we need to know if you have your games in your LaunchBox folder or in their own Roms folder. If they're in the LB folder you have a relative path, and if they're in their own folder then they are in a full path, like Derek has shown.
If they're in a full path then move all your games to your external. If they're in say, "C:\Emulators\insert system names\" and you move just the emulators folder, you can use Notepad++ to find and replace the "C:\" with your external letter, lets say "D:\". If your games are already on an external hard drive or you've just moved them to an external then on your new computer just make sure to change your external hard drive letter to the old one. If you plug in the external to your new PC it may not stay the old letter, it depends on what you have plugged in at the time. So if on the new PC it's F:\, then in windows before you start doing too much with your PC change it to D:\. You'll also want to make sure your Emulators have the same path, if they were in the full path you changed then you're good, but if they're in a completely different path then you need to double check those, but there shouldn't be too many so you can do that manually.
As per the LaunchBox folder, that is completely Portable as it never installs anything to your Windows Registry. So as long as your console names in LB never change, all the images, video, music and manuals will load just fine.