In this paper we study the behaviour of normal cones and subdifferentials with respect to two types of convergence of sets and functions: Mosco and Attouch–Wets convergences. Our analysis is devoted to proximal, Fréchet, and Mordukhovich limiting normal cones and subdifferentials. The results obtained can be seen as extensions of the Attouch theorem to the context of non-convex functions on locally uniformly convex Banach space. They also generalize, to sequences of subsmooth sets or functions, various results in the literature.