  We study the evolution of the fine-structure constant, $\alpha$, induced by
non-linear density perturbations in the context of the simplest class of
quintessence models with a non-minimal coupling to the electromagnetic field,
in which the two available free functions (potential and gauge kinetic
function) are Taylor-expanded up to linear order. We show that the results
obtained using the spherical infall model for an infinite wavelength
inhomogeneity are inconsistent with the results of a local linearized gravity
study and we argue in favour of the second approach. We also discuss recent
claims that the value of $\alpha$ inside virialised regions could be
significantly different from the background one on the basis of these findings.
