The setting: $M$ an arbitrary smooth manifold, $G$ a Lie group acting effectively and properly on $M$ by diffeomorphisms.

Motivation: when trying to figure out the homeomorphism type of the orbit space, one tries to find a "fundamental domain" $D$ that contains exactly one representative of each orbit except for a set of measure zero on which identifications may occur, in which case you have some orbits with more than one representative. You want the projection $\pi:M \rightarrow M / G$ to restrict to a quotient map on your fundamental domain so that you're sure $D / G$ is homeomorphic to $M / G$.

I have only seen good definitions for what is meant by "fundamental domain" in the case of properly discontinuous actions, and the conversation seems to focus on the free action of a discrete Lie group—see Misha's answer to Proper discontinuity and existence of a fundamental domain for example—, but I am interested in non-free actions of not-necessarily-discrete Lie groups, and researchers in my field take "fundamental domain" to be an undefined intuitive concept, so think of a "fundamental domain" in whatever way you see fit.

The question: can you think of any example of an action in this setting where you can restrict to a subset (no restrictions on the properties of the set, but the less pathological the better) $D$ that contains at least one representative of every orbit, but such that $D / G$ is not homeomorphic to $M / G$ or (equivalently?) $\pi\rvert_D$ fails to be a quotient map?

In other words, is it possible for the "fundamental domain" thinking to go wrong?

If not, can you prove a sense in which it always works?