Lipschitz-free operators $T_f\colon\mathcal{F}(M)\to\mathcal{F}(M)$ provide a natural way to linearize the dynamics of a Lipschitz self-map $f\colon M\to M$ through the associated Lipschitz-free space. In this talk, we will survey recent progress in their dynamical study, with emphasis on some new criteria to obtain hypercyclicity-related phenomena. Based on joint work with R. Ernst, Q. Menet, and A. Peris.