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We present results from MUSE spatially-resolved spectroscopy of 21 post-starburst galaxies in the centers of 8 clusters from $z\sim0.3$ to $z\sim0.4$. We measure spatially resolved star-formation histories (SFHs), the time since quenching ($t_Q$) and the fraction of stellar mass assembled in the past 1.5 Gyr ($μ_{1.5}$). The SFHs display a clear enhancement of star-formation prior to quenching for 16 out of 21 objects, with at least 10% (and up to $>50$%) of the stellar mass being assembled in the past 1.5 Gyr and $t_Q$ ranging from less than 100 Myrs to $\sim800$ Myrs. By mapping $t_Q$ and $μ_{1.5}$, we analyze the quenching patterns of the galaxies. Most galaxies in our sample have quenched their star-formation from the outside-in or show a side-to-side/irregular pattern, both consistent with quenching by ram-pressure stripping. Only three objects show an inside-out quenching pattern, all of which are at the high-mass end of our sample. At least two of them currently host an active galactic nucleus. In two post-starbursts, we identify tails of ionized gas indicating that these objects had their gas stripped by ram pressure very recently. Post-starburst features are also found in the stripped regions of galaxies undergoing ram-pressure stripping in the same clusters, confirming the link between these classes of objects. Our results point to ram-pressure stripping as the main driver of fast quenching in these environments, with active galactic nuclei playing a role at high stellar masses.