  We prove the homogenization of fully nonlinear parabolic equations with
periodic oscillating Dirichlet boundary conditions on certain general
prescribed space-time domains. It was proved in [9,10] that for elliptic
equations, the homogenized boundary data exists at boundary points with
irrational normal directions, and it is generically discontinuous elsewhere.
However for parabolic problems, on a flat moving part of the boundary, we prove
the existence of continuous homogenized boundary data $\bar{g}$. We also show
that, unlike the elliptic case, $\bar{g}$ can be discontinuous even if the
operator is rotation/reflection invariant.
