The authors consider families of non-autonomous partial differential inclusions (multivalued) with \(p\)-Laplacians with variable coefficients as differential operators, large diffusions and driven by nonlinearities of Heaviside type, i.e., corresponding to discontinuous nonlinear terms and with Neumann boundary conditions. It is proved that they generate a sequence of multivalued nonautonomous dynamical systems with a pullback attractor. It is then shown that the solutions converge to the solutions of a limit system of ordinary differential inclusions for large diffusion and when the exponents go to 2. The upper semicontinuity of pullback attractors is proved as well.