Rou,
Yes, I am curious what multi-path issues you are thinking about.
In regards to one system serving mutliple aerodromes, I would have to ponder that.
Currently, the system needs at least 3 ant, and 4 is optimal, placed as far apart as possible, but still within a certain zone. These feed the unit, which is balancing all of the GPS signals, but several other factors in the accuracy. Atmospheric conditions are considered, # of sats, etc, nothing new there.
The information from the ant is fed to a single ground unit, which then balances the data based on the location of the ground unit. The correction factor is based on this location, and the advertised accuracy of the GBAS system is based on this location.
That being said, the system accuracy is tested for the aerodrome and associated runway ends. I am not aware of any locations where the accuracy of the broadcast has been tested for points on aerodromes nearby. It was never meant to be wide area augmentation.
Internally, I am not sure of the grid size or what method the GBAS system uses in its determination relative to the geiod. That would be the issue with trying to expand the system to other aerodromes.
This not only relates to the GBAS system, but has a correlation to the particular avionics as well. If you get too far outside the grid or lookup function of the onboard GPS, accuracy will degrade or potentially disco.
Hope that helps.
EDIT: Just noting that aviation needs to get out of the 'lowest common denominator' factor when providing systems at an aerodrome. There have always been casualties of evolution.