  Non-linear electrodynamic models are re-assessed in this paper to pursue an
investigation of the kinematics of the Compton effect in a magnetic background.
Before considering specific models, we start off by presenting a general
non-linear Lagrangian built up in terms of the most general Lorentz- and
gauge-invariant combinations of the electric and magnetic fields. The extended
Maxwell-like equations and the energy-momentum tensor conservation are
presented and discussed in their generality. We next expand the fields around a
uniform and time-independent electric and magnetic backgrounds up to second
order in the propagating wave, and compute dispersion relations which account
for the effect of the external fields. We obtain thereby the refraction index
and the group velocity for the propagating radiation in different situations.
In particular, we focus on the kinematics of the Compton effect in presence of
external magnetic fields. This yields constraints that relate the derivatives
of the general Lagrangian with respect to the field invariants and the magnetic
background under consideration. We carry out our inspection by focusing on some
specific non-linear electrodynamic effective models: Hoffmann-Infeld,
Euler-Heisenberg, generalized Born-Infeld and Logarithmic.
