"The second [...] is defined by taking the fixed numerical value of the caesium frequency, ΔνCs, the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, to be 9192631770 when expressed in the unit Hz, which is equal to s−1.[1]"
That's the whole point, it does. Or, said differently, it behaves the same in whatever reference frame it's in, but observers in other reference frames can observe it being slower or faster.
Here's an example with very elementary particles, muons:
Cesium on the moon would continue to have the same frequency on the moon, but a cesium clock on the moon would run 56 microseconds fast per day relative to an identical clock on earth, because time itself moves differently.
So I'm not an expert, but this behavior actually would behave differently in different gravity wells. Due to relativity, while you would observe a cesium clock working normally on the moon, your Earth buddy would disagree with your clock because the buddy would be down in a larger gravity well.
Sortof. It would behave the same to the observer sitting next to the atomic clock no matter their local frame of reference. The point though is that we have atomic clocks on both Earth and (soon) the Moon, so we need some sort of standard for coordinating between them.
https://en.wikipedia.org/wiki/Second
so unless caesium behaves differently at different locations in the universe, this definition should be pretty uniform.