  The existing observational data on possible variations of fundamental
physical constants (FPC) confirm more or less confidently only a variability of
the fine structure constant $\alpha$ in space and time. A model construction
method is described, where variations of $\alpha$ and other FPCs (including the
gravitational constant $G$) follow from the dynamics of extra space-time
dimensions in the framework of curvature-nonlinear multidimensional theories of
gravity. An advantage of this method is a unified approach to variations of
different FPCs. A particular model explaining the observable variations of
$\alpha$ in space and time has been constructed. It comprises a FRW cosmology
with accelerated expansion, perturbed due to slightly inhomogeneous initial
data.
