I was wondering if there is some kind of Einstein's field equation for orbifolds (say semi-Riemannian of Lorentz signature if this make sense).

Here, by an orbifold I mean the "stacky" quotient of, for example, a compact Lie group $G$ acting smoothly on a manifold $M$ and with finite stabilizers...hence the stacky quotient will be (the Morita equivalence class of) its translation (Lie) groupoid $[M\times G\rightrightarrows M]$, in contrast to the "coarse" orbit space $M/G$ which is an object with singularities.

For this kind of stacky objects there is a theory of connections and curvature for $S^1$-bundles (related with Maxwell's equations, I guess)...I´m just curious if there exist more "classical" notions of curvature which allow to state Einstein´s field equation in this context...of course, if this is the case it would be great to think about its meaning from the perspective of singularities as well.