It really depends on the program you work for..
In the program I currently work for, although we have multiple bases, we are assigned an individual base and shift. Of course, we are able to cover other bases as needed also.
I can't speak for all programs, naturally, but I do believe that this is the case with most. Some HEMS programs also have personnel assigned as "float" staff who cover shortages regardless of the base location within that program. However, again, this is program specific.
That said, as a member of these services, one is also obligated at times to cover bases other than your assigned base when needed. And so, there's always the chance of moving base to base even in operations where it's uncommon.
As you mentioned in your question, there truly isn't a definite answer to this. However, I hope my opinion may have helped a little.