msumm21 said:
I'm trying to see how, when viewing a process (transition between energy levels) from a relatively moving frame, the process appears to occur slower.
As others have pointed out, it's not actually the transition rate that you are interested in (although that will decrease by the same factor and you could use it as a very poor clock). The oscillator in the clock is the EM radiation emitted by the transition.
Now here's the fun thing. In any frame except the rest frame of the atom, the emitted frequency depends on the angle at which it's emitted, presumably because the electromagnetic field of the nucleus is not spherically symmetric in those frames. Thus it's not at all clear to me that there is "a" change to the energy levels of the atom. Without going into the maths myself, I would say it isn't even certain that "energy level" is a useful concept in other frames (although others who actually understand relativistic quantum field theory may correct me on that).
It's perfectly fine to want to understand the start-to-finish process of any mechanism. However, that doesn't seem to be what you want. You seem to want a soundbite on "how it works", and I'm not really sure there is such a thing. The whole reason the lightclock gets used in relativity is that it's pretty much the only one that is trivial to analyse. Even the mass on the spring gets messy because the transform of the three-force and three-momentum is not simple.
So the point we're all trying to make is that the description of a Cs clock, even at the level of
@Dale's Doppler explanation (which is perfectly fine, but doesn't cover the photon emission at all), is a lot more complex in the moving frame. There probably isn't a soundbite summary of "why it ticks slowly", whereas with the light clock there is nothing more complex than a bit of high-school level geometry, and you can derive the full Lorentz transforms with just two of them.
Understanding that "all clocks must dilate the same" follows directly from the principle of relativity is a much more valuable insight than anything short of a complete QED description of the caesium clock.