I've got 2 lots of the stations, one with the full 77 sets and another with just a dozen so far so they definitely don't carry over.
You can pick them up and move them after they've been attuned without losing them tho.
One thing from PTS tho, that I don't know whether was fixed on not: if you have 2 of the same station with different number of sets in the same house, the one you pick up will be interchangeable - so if you have a full one out and put down an empty one then pick it up again, you might find you have the full one in your inventory instead. Not something that will come up very often but worth being aware of!