Yes, this does happen, and there were about 8 students in my year who transferred from St Andrew's. I think you may have to do an extra "honours" year or something like that before being able to transfer here, though tbh you'd be best speaking to one of your own people about that side of things. I'm not exactly sure how the application process works for you, but you'd be applying in competition with all the Cambridge medics and anyone else who wanted to get in; they look at your results for the first two years, what work experience you have, any special academic achievements or interests etc. I'm pretty sure they interview everyone who puts Cambridge down as their first choice. If you get a place, then you will join the Cambridge students at the start of their 4th year and just go through the normal clinical course with them.
Hope that helps, sorry I can't be more specific on the application process. For Cambridge undergrad students, there is a form to fill in on which you rank Oxbridge and the 5 London medical schools in order of preference, but I would imagine that you have to do something different as my understanding is that most St. Andrew's students do not come this far south! There doesn't seem to be much info on the Cambridge website on a quick search, but you can always try contacting the clinical school directly.