Measuring Light Speed Outside

Light originating from within the experiment frame of reference, travelling at high speed, is still \(c\),

 

but what about light from outside the experiment frame of reference?  If this light beam is still measured at \(c\), then there is no Doppler Effects; relativistic Doppler shift will not apply.

Does time originates from outside the moving experiment frame of reference or does the experiment scoop a 'piece of time' with it?