  In this paper we study the limit as $p\to \infty$ in the evolution problem
driven by the $p-$Laplacian with dynamical boundary conditions. We prove that
the natural energy functional associated with this problem converges to a limit
in the sense of Mosco convergence and as a consequence we obtain convergence of
the solutions to the evolution problems. For the limit problem we show an
interpretation in terms of optimal mass transportation and provide examples of
explicit solutions for some particular data.
