  We consider the Pleba{\'n}ski class of nonlinear theories of vacuum
electrodynamics, i.e., Lagrangian theories that are Lorentz invariant and gauge
invariant. Our main goal is to derive the transport law of the polarization
plane in such a theory, on an unspecified general-relativistic spacetime and
with an unspecified electromagnetic background field. To that end we start out
from an approximate-plane-harmonic-wave ansatz that takes the generation of
higher harmonics into account. By this ansatz, the electromagnetic field is
written as an asymptotic series with respect to a parameter $\alpha$, where the
limit $\alpha \to 0$ corresponds to sending the frequency to infinity. We
demonstrate that by solving the generalized Maxwell equations to zeroth and
first order with respect to $\alpha$ one gets a unique transport law for the
polarization plane along each light ray. We exemplify the general results with
the Born-Infeld theory.
