Is there a chain rule of any kind for the generalised directional derivative (of the Clarke type)? There is certainly a chain rule for the generalised gradient.

The generalised directional derivative is: $$f^\circ(x;v)=\limsup_{y \to x, t \downarrow 0} \frac{f(x+tv) - f(x)}{t}.$$

Some information about it can be found here: https://www.encyclopediaofmath.org/index.php/Clarke_generalized_derivative