In this work we consider the existence and uniqueness of variational solutions to a class of inhomogeneous evolution reaction-diffusion problems with jump discontinuous coefficients, corresponding to different reactions in each one of the phases. We discuss the cases of local and nonlocal diffusions and the convergence of the solutions of the nonlocal problems towards the solutions of the local ones when the fractional parameter s tends to 1. We also consider the possible dependence of the threshold of the discontinuity on the solution, corresponding to quasi-variational inequalities. This is a joint work with Pedro Campos.