An action of a countable group on a C*-algebra is called equivariantly Jiang-Su stable if it tensorially absorbs the trivial action on the Jiang-Su algebra. This is an important regularity property in the context of the classification of C*-dynamical systems. In this talk I will discuss joint work with Gabor Szabo, which shows that for actions of amenable groups on unital, separable, simple, nuclear, Jiang-Su stable C*-algebras, equivariant Jiang-Su stability can be deduced from an a priori weaker property called equivariant property Gamma (a dynamical version of uniform property Gamma introduced by Castillejos et al. a few years ago). More specifically, I will introduce equivariant property Gamma, explain how it allows us to deduce certain behavior of the dynamical system with respect to the uniform tracial 2-norm from behavior with respect to the individual tracial 2-norms (we call this the local-to-global principle), and how one can use this to obtain equivariant Jiang-Su stability.