Assume that $G$ is an affine reductive algebraic group (I am mostly interested in the case $GL_n$) over an algebraically closed field $K$ of characteristic zero. Assume also that $G$ acts on an affine variety $X$ with finite stabilizers. I would like to ask how does the isomorphism type $Stab_G(x)$ varies with respect to $X$. For example, for every finite group $H$ we can define $$X_H=\{x\in X| Stab_G(x)\cong H\}.$$ What can we say about the sets $X_H$? are they open? closed? constructible? Is there something that can be said about $X_H$ if for example $|H|$ is maximal or minimal among all stabilizers?

If we were talking about the dimension of $Stab_G(x)$ instead of the cardinality, then there are some semi-continuity results: the set of points $x$ for which $dim(Stab_G(x))$ is $\geq n$ is closed for example (where $n$ is some natural number). I wonder what can be said if we consider actions in which all the stabilizers are finite.