Is conditional independence of random variables preserved under diffeomorphisms?

Question

I have a general question to understand when conditional independence is preserved.

Let $(X, Y, \mathbf{Z})$ where $X$ and $Y$ are random variables and $\mathbf{Z}$ is a random vector. If $X \perp Y | \mathbf{Z}$, then under what conditions are $f(X) \perp g(Y) | h(\mathbf{Z})$ valid, where $f, g$ and $h$ are valid functions?

According to conditional independence of functions of random variables it seems that this does not hold in general.

1. However, is it true for $f, g$ and $h$ that are diffeomorphisms?
2. What is the intuition for why it does not hold in general, but it does for diffeomorphisms?
3. What are other constraints on the functions $f, g,$ and $h$ that preserve conditional independence?