Dynamical scalar fields in an effective four-dimensional field theory are naturally expected to couple to the rest of the theory's degrees of freedom, unless some new symmetry is postulated to suppress these couplings. In particular, a coupling to the electromagnetic sector will lead to spacetime variations of the fine-structure constant, α . Astrophysical tests of the space-time stability of α are therefore a powerful probe of new physics. Here we use ESPRESSO and other contemporary measurements of α , together with background cosmology data, local laboratory atomic clock and weak equivalence principle measurements, to place stringent constraints on the simplest examples of the two broad classes of varying α models: Bekenstein models and quintessence-type dark energy models, both of which are parametric extensions of the canonical Λ CDM model. In both cases, previously reported constraints are improved by more than a factor of ten. This improvement is largely due to the very strong local constraints, but astrophysical measurements can help to break degeneracies between cosmology and fundamental physics parameters.