Since the ultimate destination of the flight is Non-Schengen, all passengers boarding at Madrid have to be treated as flying to a Non-Schengen destination, and therefore go through immigration control (where they can use photo ID or passport). You can't separate two lots of pax for boarding the same plane.
At Paris, all the pax getting off are treated as Schengen arrivals and do not go through passport control.
All those arriving at LGW go through immigration since they have exited the Schengen zone.
Makes it tricky with these stopping services!