Chain rule for subdifferentials of nonconvex functions

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I have two functions: one of them $h\colon\mathbb{R}^n\to\mathcal{S}$ is smooth, but not necessarily convex, and the other $g\colon\mathcal{S}\to\mathcal{S}$ is convex, non-expansive, and not differentiable. Also, $\mathcal{S}$ is a finite-dimensional real vector space with an inner product and I suppose that the composition $g\circ h:\mathbb{R}^n\to \mathcal{S}$ is well-defined.

I will denote by $Dh(x)$ the derivative of $h$ at $x$. The B-subdifferential of $g$ at $x$ is the set $$\partial g(w)= \{W\in \mathcal{S}^* \ \colon \ \exists \{w_k\}_{k\in \mathbb{N}}\subset \mathcal{D}, \ w_k \to w, \ \nabla g(w_k)\to W\},$$ where $\mathcal{D}$ is the set in which $g$ is differentiable and $\nabla g(w_k)$ is the gradient of $g$ at $w_k$.

Is there any kind of chain rule similar to this: $$\partial (g\circ h)(x) \subseteq \bigcup_{W\in \partial g(h(x))} W[Dh(x)]$$ without any additional hypotheses, in the literature?

I saw some papers dealing with nonsmooth chain rules for $g:\mathcal{S}\to \mathbb{R}$, for set-valued mappings, Banach spaces, Asplund spaces (all by Mordukhovich), for the case $h$ is affine, and for the case $g$ and $h$ are convex (I think it was by Rockafellar). The specific cases are not useful to me, but the general cases have too many hypotheses (qualification conditions). I would be happy with results concerning the Clarke subdifferential as well.

EDIT: I only want $\subset$ actually, I don't need $=$. Also, $W[Dh(x)]$ is an abuse of notation for illustration purposes.