We use the EAGLE hydrodynamical simulation to trace the quenching history of galaxies in its 10 most massive clusters. We use two criteria to identify moments when galaxies suffer significant changes in their star formation activity: {it i)} the instantaneous star formation rate (SFR) strongest drop, $Gamma_{rm SFR}^{rm SD}$, and {it ii)} a quenching criterion based on a minimum threshold for the specific SFR of $lesssim$ 10$^{-11}rm yr^{-1}$. We find that a large fraction of galaxies ($gtrsim 60%$) suffer their $Gamma_{rm SFR}^{rm SD}$ outside the clusters R$_{200}$. This ``pre-processed population is dominated by galaxies that are either low mass and centrals or inhabit low mass hosts ($10^{10.5}$M$_{odot} lesssim$ M$_{rm host} lesssim 10^{11.0}$M$_{odot}$). The host mass distribution is bimodal, and galaxies that suffered their $Gamma_{rm SFR}^{rm SD}$ in massive hosts ($10^{13.5}rm M_{odot} lesssim M_{host} lesssim 10^{14.0}M_{odot}$) are mainly processed within the clusters. Pre-processing mainly limits the total stellar mass with which galaxies arrive in the clusters. Regarding quenching, galaxies preferentially reach this state in high-mass halos ($10^{13.5}rm M_{odot} lesssim M_{host} lesssim 10^{14.5}M_{odot}$). The small fraction of galaxies that reach the cluster already quenched has also been pre-processed, linking both criteria as different stages in the quenching process of those galaxies. For the $z=0$ satellite populations, we find a sharp rise in the fraction of quenched satellites at the time of first infall, highlighting the role played by the dense cluster environment. Interestingly, the fraction of pre-quenched galaxies rises with final cluster mass. This is a direct consequence of the hierarchical cosmological model used in these simulations.
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