There are several methods files can be linked and it's likely you may get some issues regardless of how you try to update them.
Links can be 'hard-coded' in Formulas
Hard-coded through VBA
Through the use of Power Pivot
If you know you only have formula based links then use the built in 'Edit Links' option. (see attached picture) open your file, select the DATA tab and select Edit Links then select the link and Change Source... if the option is greyed out then there are no formula links between files.
Alternatively, if you only have a few files, you could try the following steps...
Create the new directory "h:\common\excel\"
Open ALL you existing Excel Files from their current locations
Save each one to the New directory in turn, but don't close the files.
Once ALL files have been saved to their new locations, then close them down.
This should take care of hard-coded formulas. Moving linked files 'normally' updates the formulas when both the donor and recipient files are open.
You may also like to try Microsoft's Add-in tool called "Inquire", to list the links your file has with the outside world.
If your links are via Macros or VIB, you will probably have to update them manually.
In future, try to set your VBA code so it opens and saves along the lines of...
ActiveWorkbook.Path & "\" & "\" & myFile.xlsx
That way, it won't matter what the folder is the file is in.
If the links have been created using Power Pivot, unfortunately you'll need to manually update them
One final point, I see you are increasing the number of characters in your file path. Be aware that there is a limit to the number of characters you can have. Once reached, the links may not work.