  There is a growing interest in astrophysical tests of the stability of
dimensionless fundamental couplings, such as the fine-structure constant
$\alpha$, as an optimal probe of new physics. The imminent arrival of the
ESPRESSO spectrograph will soon enable significant gains in the precision and
accuracy of these tests and widen the range of theoretical models that can be
tightly constrained. Here we illustrate this by studying proposed extensions of
the Bekenstein-type models for the evolution of $\alpha$ that allow different
couplings of the scalar field to both dark matter and dark energy. We use a
combination of current astrophysical and local laboratory data (from tests with
atomic clocks) to show that these couplings are constrained to parts per
million level, with the constraints being dominated by the atomic clocks. We
also quantify the expected improvements from ESPRESSO and other future
spectrographs, and briefly discuss possible observational strategies, showing
that these facilities can improve current constraints by more than an order of
magnitude.
