When is reparameterization allowed?

[HEmile]

What are the precise conditions to allow reparameterization? It's likely not just that there is any differentiable path between a cost node and the stochastic node: A score function in between, for example, still gives an unbiased estimator! Furthermore, it should not be that there exists a differentiable path, but that all paths are differentiable: Otherwise, there could be behaviour that the reparameterization is missing.

Formalize these conditions and proof them.