In this paper, we study discrete approximations of semi-Dirichlet forms obtained by adding non-symmetric drift terms, expressed in terms of mutual energy measures, to resistance forms whose associated resistance metric spaces are compact. The main novelty is that the drift terms need not to be absolutely continuous with respect to the underlying measure. Under suitable smallness assumptions, we prove generalized Mosco convergence of the approximating forms and derive weak convergence of the associated Feller processes. We also demonstrate the applicability of our main theorem to post-critically finite self-similar fractals and describe the resulting perturbation of the approximating jump chains in the case of the Sierpiński gasket.