The origin of large-scale magnetic fields in cosmic structures and the intergalactic medium is still poorly understood. Many attempts are based on modifying electromagnetism in the very early Universe. Here, we explore the effects of non-minimal couplings of electromagnetism during later stages of th evolution of the Universe, by investigating 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's law valid on large scales as well as in the presence of non-minimal couplings to cosmological (pseudo-)scalar fields. Due to the sizable 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.