Kinematic effects of Special Relativity like time dilation and length contraction are well known.
Article in Wikipedia makes it clear:
“In the special theory of relativity, a moving clock is found to be ticking slowly with respect to the observer's clock. If Sam and Abigail are on different trains in near-lightspeed relative motion, Sam measures (by all methods of measurement) clocks on Abigail's train to be running slowly and similarly, Abigail measures clocks on Sam's train to be running slowly.”
One practical way to measure the amount of time dilation is measuring frequency of relatively moving source of radiation at points of closest approach – the Transverse Doppler Effect. That means, if Sam radiates with proper frequency f, Abigail measures frequency $f_s/\gamma$. If Sam measures frequency of Abigail, he measures $f_a/\gamma$. That means, the photon redshifts and redshifts again.
According to this logic, if Sam and Abigail exchange a photon, the photon will finally vanish. Or will Sam and Abigail will vanish themselves? Where does the energy disappear?
Please don't confuse it with relativistic Doppler redshift when emitter and absorber recede.