I will present a joint work with Romain Petrides (Université Paris Cité) where we propose a general framework to study mapping properties of critical points of functionals F(g) = F(S_g), where g runs over an open set of Riemannian metrics on a given smooth manifold, S_g is a set of eigenvalues of an elliptic operator depending on g, and F is a locally Lipschitz function. At the core of our approach is Clarke's notion of subdifferential. Our work covers well-known cases, like Laplace and Steklov eigenvalues, and provides promising perspectives in the context of metric measure spaces which I will describe at the end.