On the transitivity of the group of orbifold diffeomorphisms

Consider a connected manifold of dimension at least two and the group of compactly supported diffeomorphisms that are isotopic to the identity through a compactly supported isotopy. This group acts n-transitively: any n-tuple of points can be moved to any other n-tuple by an element of this group. The group of diffeomorphisms of an orbifold is typically not n-transitive: simple obstructions are given by isomorphism classes of isotropy groups of points. In this paper we investigate the transitivity properties of the group of compactly supported diffeomorphisms of orbifolds that are isotopic to the identity through a compactly supported isotopy. We also study an example in the category of area preserving mappings.