Atomic clocks don't use atomic decay. They used the frequency of the light emitted by a very specific energy change, within an atom, under very controlled conditions.

The frequency of light will look the same on the moon. However, an observer on earth would see a very slightly different frequency from the moon clock.

The way I understand it is that time itself is altered by gravity and/or velocity. So atomic decay that occurs on a very specific cadence in each reference frame will not occur simultaneously in 2 or more different reference frames that are not in the same gravity, moving at the same velocity. There's even a measurable though very small difference in the passage of time between sea level and high mountains due to the difference in gravity. I'm leaving a lot out and there's a bunch of math involved, but i think that's mostly correct.