  Astrophysical tests of the stability of dimensionless fundamental couplings,
such as the fine-structure constant $\alpha$ and the proton-to-electron mass
ratio $\mu$, are an optimal probe of new physics. There is a growing interest
in these tests, following indications of possible spacetime variations at the
few parts per million level. Here we make use of the latest astrophysical
measurements, combined with background cosmological observations, to obtain
improved constraints on Bekenstein-type models for the evolution of both
couplings. These are arguably the simplest models allowing for $\alpha$ and
$\mu$ variations, and are characterized by a single free dimensionless
parameter, $\zeta$, describing the coupling of the underlying dynamical degree
of freedom to the electromagnetic sector. In the former case we find that this
parameter is constrained to be $|\zeta_\alpha|<4.8\times10^{-6}$ (improving
previous constraints by a factor of 6), while in the latter (which we
quantitatively compare to astrophysical measurements for the first time) we
find $\zeta_\mu=(2.7\pm3.1)\times10^{-7}$; both of these are at the $99.7\%$
confidence level. For $\zeta_\alpha$ this constraint is about 20 times stronger
than the one obtained from local Weak Equivalence Principle tests, while for
$\zeta_\mu$ it is about 2 orders of magnitude weaker. We also discuss the
improvements on these constraints to be expected from the forthcoming ESPRESSO
and ELT-HIRES spectrographs, conservatively finding a factor around 5 for the
former and around 50 for the latter.
