Central sequences play a fundamental role in operator algebras, from property Gamma and the MacDuff property for II_1 factors to characterizations of tensorial absorption for C*-algebras, for example. Given the deep link between operator algebras and dynamics, it is natural to look for dynamical analogues of central sequence properties. This turns out to be subtle. In joint work with Grigorios Kopsacheilis, Hung-Chang Liao, and Andrea Vaccaro, a central sequence property (uniform property Gamma) is shown to be equivalent to a topological dynamical property (the small boundary property). I will discuss a body of work related to these ideas.