  We study the behaviour of solutions of linear non-autonomous parabolic
equations subject to Dirichlet or Neumann boundary conditions under
perturbation of the domain. We prove that Mosco convergence of function spaces
for non-autonomous parabolic problems is equivalent to Mosco convergence of
function spaces for the corresponding elliptic problems. As a consequence, we
obtain convergence of solutions of non-autonomous parabolic equations under
domain perturbation by variational methods using the same characterisation of
domains as in elliptic case. A similar technique can be applied to obtain
convergence of weak solutions of parabolic variational inequalities when the
underlying convex set is perturbed.
