We quantify the evolution of the spiral, S0 and elliptical fractions in galaxy clusters as a function of cluster velocity dispersion (σ) and X-ray luminosity (LX ) 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 σ between 500 and 1100 km s-1, and LX between 0.2 and 5 × 1044 erg s-1. 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 σ, 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 LX . Conversely, no correlation is observed with σ. Given that both σ and LX are tracers of the cluster mass, these results pose a challenge for current scenarios of morphological evolution in clusters.