  The origin of large-scale magnetic fields in cosmic structures and the
intergalactic medium is still poorly understood. We explore the effects of
non-minimal couplings of electromagnetism on the cosmological evolution of
currents and magnetic fields. In this context, we revisit the mildly non-linear
plasma dynamics around recombination that are known to generate weak magnetic
fields. We use the covariant approach to obtain a fully general and non-linear
evolution equation for the plasma currents and derive a generalised Ohm law
valid on large scales as well as in the presence of non-minimal couplings to
cosmological (pseudo-)scalar fields. Due to the sizeable conductivity of the
plasma and the stringent observational bounds on such couplings, we conclude
that modifications of the standard (adiabatic) evolution of magnetic fields are
severely limited in these scenarios. Even at scales well beyond a Mpc, any
departure from flux freezing behaviour is inhibited.
