  Astrophysical tests of the stability of fundamental couplings, such as the
fine-structure constant $\alpha$, are a powerful probe of new physics. Recently
these measurements, combined with local atomic clock tests and Type Ia
supernova and Hubble parameter data, were used to constrain the simplest class
of dynamical dark energy models where the same degree of freedom is assumed to
provide both the dark energy and (through a dimensionless coupling, $\zeta$, to
the electromagnetic sector) the $\alpha$ variation. One caveat of these
analyses was that it was based on fiducial models where the dark energy
equation of state was described by a single parameter (effectively its present
day value, $w_0$). Here we relax this assumption and study broader dark energy
model classes, including the Chevallier-Polarski-Linder and Early Dark Energy
parametrizations. Even in these extended cases we find that the current data
constrains the coupling $\zeta$ at the $10^{-6}$ level and $w_0$ to a few
percent (marginalizing over other parameters), thus confirming the robustness
of earlier analyses. On the other hand, the additional parameters are typically
not well constrained. We also highlight the implications of our results for
constraints on violations of the Weak Equivalence Principle and improvements to
be expected from forthcoming measurements with high-resolution ultra-stable
spectrographs.
