Galaxy clusters are the largest gravitationally bound structures in the Universe. Their masses are dominated by dark matter ($\sim$85\% of the mass) with stars representing 1-4\% of their masses. A hot, X-ray emitting gas called the intracluster medium makes most of their baryonic mass. The presence of this gas and of numerous neighbouring galaxies prematurely stop the star formation in clusters. In other terms, more galaxies in clusters are passive than in the general population of galaxies. This effect is mass and position-dependant: high-mass galaxies are more likely to be passive than less massive ones; galaxies inhabiting the cluster core are also less likely to form stars than those in the outskirts. The fraction of passive galaxies is greater in local clusters than in high-redshift ones, because they had more time to evolve. Much is unknown about the cessation of star formation, called quenching, in clusters. Thus, although many examples of infalling galaxies being stripped of their gas have been reported for low-mass galaxies, it is unclear if the most massive members became quenched before or after they become cluster members. The relationship between quenching and the cluster mass is also poorly understood. Despite the variety of methods devised to find clusters of galaxies, most of what we know about quenching in $z\gtrsim 1$ clusters was discovered with optically/infrared-selected cluster samples (clusters found as overdensities of galaxies), or samples of mixed origin. Yet, there is tentative evidence that optically/infrared-selected samples are biased toward having more passive galaxies than those that were X-ray selected. In the present dissertation, quenching is explored in X-ray selected cluster samples. A sample of high-redshift, low-mass galaxy clusters is built by finding galaxy overdensities coincident with sources of extended X-ray emission. A photometry-based analysis reveals that the fraction of quenched galaxies in these clusters is very variable. Moreover, the brightest cluster galaxies are also diverse. Yet, for all the information that photometry can provide, this sample candidate clusters need to be confirmed with spectroscopy. Spectroscopic observations obtained for four candidate clusters are reduced and analysed. The results show that three of them are clusters, the fourth candidate being a superposition of structures. Member spectra are examined to infer their star formation history, and the results shows the existence of an intermediary population of galaxies, where an old stellar population coexists with weak star formation. Finally, the galaxies of a $z=1.98$ X-ray selected cluster, XLSSC 122 are investigated in detail. Photometric data in 12 bands are organized to perform spectral energy distribution fittings, a technique that allows a simplified reconstitution of the history of the star formation. Results show that the members were formed at diverse epochs, the oldest being about 2.5 Gyrs old. Simulations drawn from the Multi Dark Planck 2 are used to infer the mass-scale of the cluster when the oldest galaxies were formed, something that has never been done before. The oldest galaxies were probably formed when XLSSC 122 had accreted $<$10\% of its $z=1.98$ mass, i.e. the mass-scale of a galaxy group. / Graduate