Given a symplectic manifold $X$ with nice $G$ action, equivariant momentmap $\mu$ and invariant $\chi \in \mathfrak{g}^*$ which is a regular value of $\mu$. There are two ways to form the Hamiltonian reduction. What one usually does is take the levelset $X_\chi=\mu^{-1}(\chi)$ and quotient out the group $X_\chi/G$. However one could also quotient out $G$ first and then define $(X/G)_\chi$ to be the set of all points represented by elements of $\mu^{-1}(\chi)$.

I have sometimes heard that these two procedures are equal. Is this true, or more precisely in what situations is it true? Is this written down somewhere? Are there counterexamples one should have in mind?

I am also interested in settings where one replaces spaces by possibly noncommutative Poisson algebras.