  We study the cosmological evolution of the fine structure constant, $\alpha$,
and the proton-to-electron mass ratio, $\mu=m_p/m_e$, in the context of a
generic class of models where the gauge kinetic function is a linear function
of a quintessence-type real scalar field, $\phi$, described by a Lagrangian
with a standard kinetic term and a scalar field potential, $V(\phi)$. We
further assume that the scalar field potential is a monotonic function of
$\phi$ and that the scalar field is always rolling down the potential. We show
that, for this class of models, low-redshift constrains on the evolution of
$\alpha$ and $\mu$ can provide very stringent limits on the corresponding
variations at high-redshift. We also demonstrate that these limits may be
relaxed by considering more general models for the dynamics of $\alpha$ and
$\mu$. However, in this case, the ability to reconstruct the evolution of the
dark energy equation of state using varying couplings could be seriously
compromised.
