Recall the 'law of the unconscious statistician' (wikipedia), namely that one can calculate the expectation of the transform $g(X)$ of a random variable $X$, given the transformation and the probability density function $f$ of the original random variable. If one is being fancy, then this is a statement about pushforward measures.
But I would like to know if one has a similarly simple result about calculating the higher central moments. If there is no closed form, then I should say I would be happy with the first four as a compromise.
To be concrete, I have a smooth function $g:\mathbb{R}^2 \to [0,1]$, and $X$ is bivariate normal (so no worries about moments existing).