There may be other answers but I think you just need standard file updating like you would for any file. Just an update program to see what files are newer on the server and copy it to the station if needed. As far as what records are new you could add a field that is a date/time stamp with auto-enter of modification date and just show the mostly recently updated records at the top of your list.
If you are simply copying complete files from the server to your various computers at home that is one problem. But if you are updating records on the server (or its clients) and on various computers at home and you want them all to synchronise their records then I think you are in for a very big headache.
If you are going to follow Njem's advice about synchronising complete files then you must have web access. Why not connect your home computers to the FM Server remotely and then all clients are using the same server file and no synchronisation is necessary?