  In this paper, we study existence, uniqueness and asymptotic behavior of the
Laplace equation with dynamical boundary conditions on regular non-cylindrical
domains. We write the problem as a non-autonomous Dirichlet-to-Neumann operator
and use form methods in a more general framework to accomplish our goal. A
class of non-autonomous elliptic problems with dynamical boundary conditions on
Lipschitz domains is also considered in this same context.
