  We consider spin systems with nearest-neighbor interactions on an $n$-vertex
$d$-dimensional cube of the integer lattice graph $\mathbb{Z}^d$. We study the
effects that exponential decay with distance of spin correlations, specifically
the strong spatial mixing condition (SSM), has on the rate of convergence to
equilibrium distribution of non-local Markov chains. We prove that SSM implies
$O(\log n)$ mixing of a block dynamics whose steps can be implemented
efficiently. We then develop a methodology, consisting of several new
comparison inequalities concerning various block dynamics, that allow us to
extend this result to other non-local dynamics. As a first application of our
method we prove that, if SSM holds, then the relaxation time (i.e., the inverse
spectral gap) of general block dynamics is $O(r)$, where $r$ is the number of
blocks. A second application of our technology concerns the Swendsen-Wang
dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies
an $O(1)$ bound for the relaxation time. As a by-product of this implication we
observe that the relaxation time of the Swendsen-Wang dynamics in square boxes
of $\mathbb{Z}^2$ is $O(1)$ throughout the subcritical regime of the $q$-state
Potts model, for all $q \ge 2$. We also prove that for monotone spin systems
SSM implies that the mixing time of systematic scan dynamics is $O(\log n (\log
\log n)^2)$. Systematic scan dynamics are widely employed in practice but have
proved hard to analyze. Our proofs use a variety of techniques for the analysis
of Markov chains including coupling, functional analysis and linear algebra.
