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The environments where galaxies reside crucially shape their star formationhistories. We investigate a large sample of 1626 cluster galaxies locatedwithin 105 galaxy clusters spanning a large range in redshift ($0.26 < z <1.13)$. The galaxy clusters are massive (M$_{500} \gtrsim2\times10^{14}$M$_{\odot}$), and are uniformly selected from the SPT and ACTSunyaev-Zel'dovich (SZ) surveys. With spectra in-hand for thousands of clustermembers, we use galaxies' position in projected phase space as a proxy fortheir in-fall times, which provides a more robust measurement of environmentthan quantities such as projected cluster-centric radius. We find clearevidence for a gradual age increase of the galaxy's mean stellar populations($\sim$ 0.71 $\pm$ 0.4 Gyr based on a 4000 $\r{A}$ break, $\rm D_{\rm n}4000$)with the time spent in the cluster environment. This environmental quenchingeffect is found regardless of galaxy luminosity (faint or bright) and redshift(low-$z$ or high-$z$), although the exact stellar age of galaxies depends onboth parameters at fixed environmental effects. Such a systematic increase of$\rm D_{\rm n}4000$ with in-fall proxy would suggest that galaxies that wereaccreted into hosts earlier were quenched earlier, due to longer exposure toenvironmental effects such as ram pressure stripping and starvation. Comparedto the typical dynamical time scales of $1-3$ Gyr of cluster galaxies, therelatively small age increase ($\sim$ 0.71 $\pm$ 0.4 Gyr) found in our samplegalaxies seems to suggest that a slow environmental process such as starvationis the dominant quenching pathway. Our results provide new insights intoenvironmental quenching effects spanning a large range in cosmic time ($\sim5.2$ Gyr, $z=0.26$--1.13) and demonstrate the power of using akinematically-derived in-fall time proxy.

A Gradual Decline of Star Formation since Cluster In-fall: New Kinematic Insights into Environmental Quenching at 0.3 $< z <$ 1.1