  Using a gauge-invariant formalism we derive and solve the perturbed
cosmological equations for the BSBM theory of varying fine structure
'constant'. We calculate the time evolution of inhomogeneous perturbations of
the fine structure constant, $\frac{\delta \alpha}{\alpha}$ on small and large
scales with respect to the Hubble radius. In a radiation-dominated universe
small inhomogeneities in $\alpha $ will decrease on large scales but on scales
smaller than the Hubble radius they will undergo stable oscillations. In a
dust-dominated universe small inhomogeneous perturbations in $\alpha $ will
become constant on large scales and on small scales they will increase as
$t^{2/3}$, and $\frac{\delta \alpha}{\alpha}$ will track $% \frac{\delta \rho
_{m}}{\rho_{m}}$ . If the expansion accelerates, as in the case of a $\Lambda $
or quintessence-dominated phase, inhomogeneities in $\alpha $ will decrease on
both large and small scales. The amplitude of perturbations in $\alpha $ will
be much smaller than that of matter or radiation perturbations. We also present
a numerical study of the non-linear evolution of spherical inhomogeneities in
radiation and dust universes by means of a comparison between the evolution of
flat and closed Friedmann models with time-varying $\alpha .$Various
limitations of these simple models are also discussed.
