Yes there really is quite a bit of moving about.
In my DD's year at her old school 4 kids moved out of London mid-year and were quickly replaced with 4 new kids all from other local preps (including both Newton and Thomas's). Then 4 more left at the end of the year all heading to other local independent schools. 2 more have since joined, again from local preps.
When we were looking to move her, I called around all the Battersea/Clapham/Balham schools and the only one that didn't either have a space or expect one to become available soon was Hornsby House.
If you do decide to move then you do have to contend with some form of selection process but having been through most of them now between my two (
the ones for the local preps we've been talking about (inc. Thomas's) aren't a big deal. Usually your child will just join in with normal classes being given the odd separate activity to complete alone so the teacher can gauge what they can/ can't do and whether they'd fit in with the class.
We ended up with 3 places to choose from.
It's slightly different if you are moving at a time that is a formal entry point for the school you want them to go to (e.g. 7+ to JAPS/JAGS, Alleyns, DC, DPL, KCJS, Westminster, Colet Court, Wimbledon/Putney...) in which case you have to contend with full on written exams.
SCP is somewhere in the middle with written maths and English tests and interview with the Head but done in a more gentle way. Newton is similar.
Hopefully you won't need to move but it's not the end of the world if you do and quite common as the kids get older and you have more of an idea of their temperament, interests, academics etc...