Hi there, I work for an ad agency and interface between the creative dept and technology (experience on both sides). I have a bit of experience with this, but I'm not a technical expert, so please keep that in mind. Over the 19 years I've been here, we have had to migrate our data multiple times to a new server, including InDesign files and their links in various directories. Everytime we do this the links break. The issue, as far as I'm aware, is that InDesign uses the entire directory path to find the location of the links. If anything in the path changes, including just the top directory level like the name of a new server, the links break. It's always painful, but luckily only needs to be done once per file, then they are back on their way.
We are having a weird issue where links are breaking immediately after saving a doc to the server and closing it. They link, save, close, open it again, and the links are broken. As a possible band-aid for this until we figure out what is going on (could be server related), I found this utility, which is only $4.99. I haven't tried it yet, but could be helpful for your users. http://www.drscripto.com/?p=8
Because a move from one server to a different I have decided to package any project for future use. It is also important as the danger would be huge that someone else edits a linked file without knowing that this file has been used in that form in a different file.
Therefore I recommend to you to package any INDD file or at least INDB files, where these INDD files belong to, before moving to the new server, after packaging you can move the whole package.
There are scripts available or plugins, like those from Zevrix, which allow mass conversion and packaging of multiple files. I would recommend buying a licence and use this.