Move a 'tracking to ' camera along its local Z axis

https://blenderartists.org/uploads/default/original/4X/f/1/3/f137d06b89ab323e83032d6cecf7fd4a98f4db3b.mp4 Hi all , I’m new to Blender and I’m facing a little problem of axis : I have a camera which is contraint to an empty ( with a ‘child to’ constraint …so to make a 'turn arround ’ stuff ) , and the cam has another constraint ‘Track to’ to aim at another object target . Now if I want to move along the local Z axis of the cam , it moves it along its hidden Z axis which is the one before the Track to constraint is apllied to : here a quick screen vid :

Is it the correct result intended ? I’m a bit lost here . I’d have excpected a ‘track to’ Z move .
here my test file : local Z move.blend (547.6 KB)