As I understand it, the claim is, starting with frame A:Yes. In a way, that is my point.
1) Do a Lorentz boost along in an arbitrary direction - call this frame B.
2) Do another Lorentz boost in an arbitrary direction - call this frame C.
3) The transformation from B to C is not a Lorentz transformation?
3b) Alternatively, the motion of the origin of frame B, as measured in frame C, is not constant velocity (speed and direction)?