Facilities abilities are not shared across bases, only copies in a base count.
You will have to build a new copy of the required facility for whatever you want to produce. Still for alcohol products, extractor for cave hunt and hellirum distilling, workshop for some general production, industrial printer for some mid-early latgame stuff, ect
There are some buildings like the library and mess hall that you need for tech unlocks that only need to be in whatever base is doing the research.
Generally speaking you will specialize each base to some task, the first base usually ends up being the main research base. Then a workshop base with all the production buildings, lots of living spaces for runts, some storage/slaves and a single hanger. Then a training base filled with dojo/spas and voodoo schools. A farm base to house your plantations.
The rest of the base slots you use to house your aircraft fleet, and for radar coverage.