TAI makes sense on Earth, but when it comes to separate cosmic bodies (like Moon base), my mind does not work well enough to understand how it would work, when

TAI makes sense on Earth, but when it comes to separate cosmic bodies (like Moon base), my mind does not work well enough to understand how it would work, when time physically is different from the Earth time and even might change if we're talking about accelerating rockets. And how one would design a software and protocols exchanging instant values between those bodies.