Our observation is that there is far more movement with girls than with boys, both because by sixteen girls can feel that an all girls school is too restrictive (my DNieces wanted to but were not allowed to leave their rather dull but posh home counties girls boarding precisely because their parents wanted them in a secure, safe environment) and because more boys schools offer co-ed sixth forms.
That said there is always a small amount of movement amongst the boys schools (and from state to private and vice versa.) Sometimes it is about fit. DD moved from a London independent co-ed to an independent boys school that took girls in the sixth form, both for an easier journey, but also because she was a bit "meh" about the first school. Looking back almost her entire friendship group at the first school gained Oxbridge places, she did not. She loved the second school and she would 100% make the move again.
Of the movement around Winchester, Eton, St Pauls and Westminster, and we have seen some, though none down to academic performance. There may be a fall off in academic achievement but this is usually triggered by some form of breakdown in the relationship with the school, often behavioural, perhaps disengagement, and one which seems to have been down to conflict due to a school not meeting high parental demands expectations. When something has gone wrong which does not impact on academic potential, schools appear to work together to facilitate a fresh start. Moves to state, whatever the parents say, are often precipitated by behaviour, drugs being the most likely. Boarding schools, in particular, fear contagion.
I don't think we ever saw anyone being asked to leave Westminster solely on academic performance grounds. There was usually something else and the numbers were small. I would assume Winchester is similar.