  The correspondence between the isoscalar monopole (IS0) transition strengths
and $\alpha$ inelastic cross sections, the $B({\rm IS0})$-$(\alpha,\alpha')$
correspondence, is investigated for $^{24}$Mg($\alpha,\alpha'$) at 130 and 386
MeV. We adopt a microscopic coupled-channel reaction framework to link
structural inputs, diagonal and transition densities, for $^{24}$Mg obtained
with antisymmetrized molecular dynamics to the ($\alpha,\alpha'$) cross
sections. We aim at clarifying how the $B({\rm IS0})$-$(\alpha,\alpha')$
correspondence is affected by the nuclear distortion, the in-medium
modification to the nucleon-nucleon effective interaction in the scattering
process, and the coupled-channels effect. It is found that these effects are
significant and the explanation of the $B({\rm IS0})$-$(\alpha,\alpha')$
correspondence in the plane wave limit with the long-wavelength approximation,
which is often used, makes no sense. Nevertheless, the $B({\rm
IS0})$-$(\alpha,\alpha')$ correspondence tends to remain because of a strong
constraint on the transition densities between the ground state and the $0^+$
excited states. The correspondence is found to hold at 386 MeV with an error of
about 20%-30%, while it is seriously stained at 130 MeV mainly by the strong
nuclear distortion. It is also found that when a $0^+$ state that has a
different structure from a simple $\alpha$ cluster state is considered, the
$B({\rm IS0})$-$(\alpha,\alpha')$ correspondence becomes less valid. For a
quantitative discussion on the $\alpha$ clustering in $0^+$ excited states of
nuclei, a microscopic description of both the structure and reaction parts will
be necessary.
