To answer the question in the thread title (and not respond to the details about the OP's specific trip involving CX), it depends on the airlines that are involved. For instance, DL will not let you change the FF number on an itinerary (this might apply only to DL itineraries or itineraries that are ticketed on DL 006- stock) after travel has begun, which technically would mean after you have boarded the first segment (and the GA has closed out the flight, the flight has departed and not diverted back to the same airport, etc.).