  Let $\mathcal{H}$ be a stratum of translation surfaces with at least two
singularities, let $m_{\mathcal{H}}$ denote the Masur-Veech measure on
$\mathcal{H}$, and let $Z_0$ be a flow on $(\mathcal{H}, m_{\mathcal{H}})$
obtained by integrating a Rel vector field. We prove that $Z_0$ is mixing of
all orders, and in particular is ergodic. We also characterize the ergodicity
of flows defined by Rel vector field, for more general spaces $(\mathcal{L},
m_{\mathcal{L}})$, where $\mathcal{L} \subset \mathcal{H}$ is an orbit-closure
for the action of $G = \mathrm{SL}_2(\mathbb{R})$ (i.e., an affine invariant
subvariety) and $m_{\mathcal{L}}$ is the natural measure. Our results are
conditional on a forthcoming measure classification result of Brown, Eskin,
Filip and Rodriguez-Hertz.We also prove that the entropy of the action of $Z_0$
on $(\mathcal{L}, m_{\mathcal{L})$ has zero entropy.
