  We consider a family of stochastic models of evolving two-dimensional Young
diagrams, given in terms of certain energies, with Gibbs invariant measures.
`Static' scaling limits of the shape functions, under these Gibbs measures,
have been shown by several over the years. The purpose of this article is to
study corresponding `dynamical' limits of which less is understood. We show
that the hydrodynamic scaling limits of the diagram shape functions may be
described by different types parabolic PDEs, depending on the energy structure.
