On each computer MAP the external Networked drive to the exact same drive letter.
If on one computer the network drive is assigned the letter H and on another computer that same network drive is seen as letter S then when you move the catalog file from PC 1 to PC 2 the catalog is looking for the images on drive H and not S. Of course this is on Windows. If you are using a Mac that uses drive names and not letters make sure the network drive is Mounted before you open Lightroom.
Thanks but this is the right answer to the wrong question. I have no problem with the photos, it's the saved locations (i.e. named lat/lon locations, see the Map functions) that are hidden away in the file I indicated. By the way Lightroom does not allow you to put the catalog file(s) on a networked drive, has to be mounted on the same computer as Lightroom so your answer does not apply to networked drives.
Oh Ok. Right no catalog on a network drive but you can store your images on one.
I think it would be fine to copy that other file over along with the catalog.
Just like you copy the catalog from computer to computer, you need to copy the user templates from your old computer to your new one.
While the catalog can be easily located in any directory you want, the templates are expected to be found in certain folders thus needing you to copy them to the expected place on the new computer.