I'm working on a problem where I have a (highly) non-transitive Lie group action on a manifold, and I am trying to deduce the geometric structure of the quotient space. I've been looking at some books like Orbifolds and Stringy Topology as well as Lie Groupoids and Lie Algebroids in Differential Geometry. Most of the theorems in these books seem geared towards transitive group actions, and I get the impression that there isn't too much that's known about non-transitive actions. Another problem is that many points in the manifold do not have finite isotropy groups. If they did, I would be able to get an orbifold structure pretty easily.
Additionally, it might be helpful to know if there are simple criteria to check when a map $f: K \to N$ from an orbifold $K$ to an arbitrary set $N$ is a diffeomorphism (in my case, $f$ is a bijection that restricts to a diffeomorphism on a subspace of $K$ whose image is a manifold).
Is there any good literature/research out there that discusses non-transitive actions in this context?