Agree with the others, switching bikes not a problem. I have several bikes and have switched individual saddles from one to another on several occasions without any problems.
But, if you were to switch from a bike with an upright position to a bike with more forward lean (more aerodynamic position) or the other way around, you might find a problem. I make that comment because I find that I can use a wider saddle when I sit more upright, but my touring bikes with drop bars need a narrower saddle, especially when I use the drops.
As fietsbob noted above, seattube angle may be different. On some of my bikes my Brooks is as far back as the seatpost clamp allows, but I have a different bike with a shallower sloping seattube, on that one I have the Brooks farther forward.