  We show that the inclusion of effects beyond Migdal's limit in the
electron-phonon interaction naturally leads to an isotope effect for the
effective mass $m^{*}$ of the charge carriers even much before reaching the
polaron limit. This is the situation already considered in our approach to
nonadiabatic superconductivity. Such a result provides a scenario different
from the polaronic one for the interpretation of the recently observed isotope
effect on $m^{*}$ in YBa$_{2}$Cu$_{3}$O$_{6+x}$ and
La$_{2-x}$Sr$_{x}$CuO$_{4}$.
