We study the transition probabilities of a class of degenerate diffusions arising as models for gene frequencies in population genetics. The processes we consider are a generalization of the classical Wright-Fisher model, and they are defined through their infinitesimal generator, which is a boundary-degenerate operator defined on compact manifolds with corners, of which simplices and convex polyhedra are particular examples. Under suitable conditions, we find that the transition probabilities have a singular structure that described the absorbing and reflecting behavior of the underlying diffusion on the boundary components of the compact manifold with corners. This is joint work with Charles Epstein.