We investigate the development of the colour-magnitude re- lation (CMR) of cluster galaxies. This study is carried out using a semi- analytic model of galaxy formation and evolution coupled to a sample of simulated galaxy clusters of different masses, reinforcing the conclusions reached by Jimenez et al. (2009). We compare both simulated and obeserved CMRs in different colour-magnitude planes, finding a very good agreement in all cases. This indicates that model parameters are correctly tuned, giving accurate values of the main properties of galaxies for further use in our study. In the present work, we perform a statistical analysis of the relative contribution to the stellar mass and metallicity of galaxies along the CMR by the different processes involved in their formation and evolution (i.e. quiescent star formation, disc instability events and galaxy mergers). Our results show that a mix of minor and major dry mergers at low redshifts is relevant in the evolution of the most luminous galaxies in the CMR. These processes contribute with low metallicity stars to the remnant galaxies, thus increasing the galaxy masses without significantly altering their colours. These results are found for all simulated clusters, supporting the idea of the universality of the CMR in agreement with observational results.