Let's imagine that I have two standing clocks and I sit in the middle. Sending two rays of light to both of them I can be sure that in my frame of reference they are synchronized.
If now I put them in motion, Lorentz transformations say that I do not see them showing the same time anymore. This cannot be due to time dilation effects since they move at the same velocity w.r.t. me.
Besides maths, how would you physically explain this de-synchronization phenomenon? It is something happening when I put them in motion (i.e. when I accelerate them)?