On the $n$-transitivity of the group of equivariant diffeomorphisms

Let $G$ be a Lie group and let $M$ be a proper smooth $G$-manifold. If $M$ is connected and $\dim(M)\geq 2$, the group of diffeomorphisms of $M$, that are isotopic to the identity through a compactly supported isotopy, acts $n$-transitively on $M$, for any $n$. In this paper, we prove a version of the $n$-transitivity result for the group of equivariant diffeomorphisms of $M$. As a corollary we obtain a result concerning diffeomorphisms of the orbit space $M/G$. A special case of the result for orbit spaces gives an $n$-transitivity result for orbifold diffeomorphisms that was earlier proved by F. Pasquotto and T. O. Rot.