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ABSTRACT The Reionization Cluster Survey imaged 41 galaxy clusters with the Hubble Space Telescope (HST), in order to detect lensed and high-redshift galaxies. Each cluster was imaged to about 26.5 AB mag in three optical and four near-infrared bands, taken in two distinct visits separated by varying time intervals. We make use of the multiple near-infrared epochs to search for transient sources in the cluster fields, with the primary motivation of building statistics for bright caustic crossing events in gravitational arcs. Over the whole sample, we do not find any significant (≳5σ) caustic crossing events, in line with expectations from semi-analytical calculations but in contrast to what may be naively expected from previous detections of some bright events or from deeper transient surveys that do find high rates of such events. Nevertheless, we find six prominent supernova (SN) candidates over the 41 fields: three of them were previously reported and three are new ones reported here for the first time. Out of the six candidates, four are likely core-collapse SNe – three in cluster galaxies, and among which only one was known before, and one slightly behind the cluster at z ∼ 0.6–0.7. The other two are likely Ia – both of them previously known, one probably in a cluster galaxy and one behind it at z ≃ 2. Our study supplies empirical bounds for the rate of caustic crossing events in galaxy cluster fields to typical HST magnitudes, and lays the groundwork for a future SN rate study.