Set of $x$ such that $h \mapsto hx$ is proper

Question

Let $X$ be a locally compact second countable space, and $G$ a locally compact second countable group wich operates continuously on $X$. If $x \in X$, let $\rho_x : g \mapsto gx$. I would like to know if anything can be said about the topology of the set of $x \in X$ such that $\rho_x$ is proper (that is, for any compact subset $K$ of $X$, $\rho_x^{-1}(K)$ is compact). I know (by looking at examples) it may happen that this set (let us call it $D$) is neither open nor closed. For example is $D$ Borel?