  We study the detailed evolution of the fine-structure constant $\alpha$ in
the string-inspired runaway dilaton class of models of Damour, Piazza and
Veneziano. We provide constraints on this scenario using the most recent
$\alpha$ measurements and discuss ways to distinguish it from alternative
models for varying $\alpha$. For model parameters which saturate bounds from
current observations, the redshift drift signal can differ considerably from
that of the canonical $\Lambda$CDM paradigm at high redshifts. Measurements of
this signal by the forthcoming European Extremely Large Telescope (E-ELT),
together with more sensitive $\alpha$ measurements, will thus dramatically
constrain these scenarios.
