We use recent astrophysical and local tests of the stability of the fine-structure constant , \alpha , to constrain a particular phenomenological but physically motivated class of models in which the dark energy equation of state can undergo a rapid transition at low redshifts , perhaps associated with the onset of the acceleration phase . We set constraints on the phenomenological parameters describing such possible transitions , in particular improving previous constraints ( which used only cosmological data ) on the present-day value of the dark energy equation of state in these models . We specifically quantify how these constraints are improved by the addition of the \alpha measurements . We find no evidence for a transition associated with the onset of acceleration . In this model the \alpha measurements lead to a bound on Weak Equivalence Principle violations of \eta < 4 \times 10 ^ { -15 } ( at 68.3 \% confidence level ) , improving on the recent MICROSCOPE bound by about a factor of three .