I've worked with the impression that it was less one soul replacing or pushing out another, but one (or more) being added to the mixture, allowing for the experiences and perspectives of the new soul to affect and teach the older ones (and vice versa) as the person lives their life, resulting in a single entity (with the experiences and memories of several) over the years.
I've also gone with the assumption that it requires training to be able to sort all of these things out in your head and to be able to make use of them. Every body (space intentional) is different, and so one has to go through the training each time in order to 'unlock' controlled access to those other minds and lives.
I've actually been playing around with this with Morwen for the last year or so, but only a few people actually know that's what's going on IC - Ava, Eva (Punx), Reppy, Sakaane, maybe Bataav (depending on some unfinished RP) and Mammal (still need to do that RP), for starters.
If people have noticed her having sudden, unexplained mood swings or demeanor/personality shifts over the last year or so, there's more to it than meets the eye - she only recently got instruction on how to make use of this part of her life, so now these shifts are often (but not always) deliberate and purposeful. In the past those shifts have been very involuntary and very unsettling/irritating to her, precisely because they weren't under her control and often happened in times and places that they were anything but helpful.