We employ the Monge-Kantorovich mass transfer theory to study the existence of solutions for a large class of parabolic partial differential equations. We deal with nonhomogeneous nonlinear diffusion problems (of Fokker-Planck type) with time-dependent coefficients. This work greatly extends the applicability of known techniques based on constructing weak solutions by approximation with time-interpolants of minimizers arising from Wasserstein-type implicit schemes. It also generalizes previous results of the authors, where proofs of convergence in the case of a right-hand side in the equation is given by these methods. To prove the existence of weak solutions we establish an interesting maximum principle for such equations. This involves comparison with the solution for the corresponding homogeneous, time-independent equation.