An important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in particular, the classical notions of weak and strong spatial mixing, which have been used to show the best known mixing time bounds in the high-temperature regime for the Glauber dynamics for the Ising and Potts models. Glauber dynamics for the random-cluster model does not naturally fit into this spin systems framework because its transition rules are not local. In this paper, we present various implications between weak spatial mixing, strong spatial mixing, and the newer notion of spatial mixing within a phase, and mixing time bounds for the random-cluster dynamics in finite subsets of $\mathbb Z^d$ for general $d\ge 2$. These imply a host of new results, including optimal $O(N\log N)$ mixing for the random cluster dynamics on torii and boxes on $N$ vertices in $\mathbb Z^d$ at all high temperatures and at sufficiently low temperatures, and for large values of $q$ quasi-polynomial (or quasi-linear when $d=2$) mixing time bounds from random phase initializations on torii at the critical point (where by contrast the mixing time from worst-case initializations is exponentially large). In the same parameter regimes, these results translate to fast sampling algorithms for the Potts model on $\mathbb Z^d$ for general $d$.