This paper is devoted to the derivation of the Spitzer-Härm limit from the coupled system of PDEs describing the evolution of charged particles and electromagnetic fields. We identify a relevant asymptotic regime which leads to a nonlinear diffusion equation for the electron temperature. Then, we discuss some intermediate models, which remain of hydrodynamic nature but involve a nonlocal coupling through integral or pseudodifferential operators. In particular, we exhibit important mathematical properties of the so-called Schurtz-Nicolaï model like the well-posedness and the maximum principle. We also design numerical schemes for the nonlocal models and analyze their consistency and stability properties.