  We quantify the evolution of the spiral, S0 and elliptical fractions in
galaxy clusters as a function of cluster velocity dispersion ($\sigma$) and
X-ray luminosity ($L_X$) using a new database of 72 nearby clusters from the
WIde-Field Nearby Galaxy-cluster Survey (WINGS) combined with literature data
at $z=0.5-1.2$. Most WINGS clusters have $\sigma$ between 500 and 1100 $\rm km
s^{-1}$, and $L_X$ between 0.2 and $5 \times 10^{44} \rm erg/s$. The S0
fraction in clusters is known to increase with time at the expense of the
spiral population. We find that the spiral and S0 fractions have evolved more
strongly in lower $\sigma$, less massive clusters, while we confirm that the
proportion of ellipticals has remained unchanged. Our results demonstrate that
morphological evolution since $z=1$ is not confined to massive clusters, but is
actually more pronounced in low mass clusters, and therefore must originate
either from secular (intrinsic) evolution and/or from environmental mechanisms
that act preferentially in low-mass environments, or both in low- and high-mass
systems. We also find that the evolution of the spiral fraction perfectly
mirrors the evolution of the fraction of star-forming galaxies. Interestingly,
at low-z the spiral fraction anticorrelates with $L_X$. Conversely, no
correlation is observed with $\sigma$. Given that both $\sigma$ and $L_X$ are
tracers of the cluster mass, these results pose a challenge for current
scenarios of morphological evolution in clusters.
