We study the $r$-band luminosity function (LF) faint end of galaxies in clusters and their environment, its mass dependence and evolution over cosmic time up to $z=2.8$. We post-processed data from the C-EAGLE simulation project, formed by 30 high-resolution simulations of galaxy clusters, computing galaxy luminosities in different bands from UV to IR, using the E-MILES stellar spectra library. The C-EAGLE numerical resolution allows us to study galaxies with a $\log(M_star)>8.2$. The (stacked) LFs inside $r_{200}$ are already well defined at z=2.8, with a well populated faint end. Overall, the LFs faint-end shows relatively little time evolution and only a modest dependence on the cluster mass. The clusters cores are rich in dwarf galaxies at all times, with their LF's faint-end constantly steeper than $\alpha=-1.28\pm 0.04$, although it does not reach the halo mass function slope. Remarkably, all the clusters develop a faint-end upturn at $z\lesssim 0.35$, best modelled with a double Schechter function, with the clusters having $\log (M_{200}/\Msun)<14.5$ being the first to develop it. We find that the LF mass dependence is due to the clusters' different evolution. The galaxies in the low mass clusters evolve faster, and are dominated by passive evolution, without significant mergers of dwarfs onto massive galaxies after $z=0.5$, and their luminosity fades with time. The massive clusters experience the vast majority of mergers and galaxy accretion down to z=0.15, confirming the scenario where the most massive clusters are the youngest and the least virialized.