  The environments where galaxies reside crucially shape their star formation
histories. We investigate a large sample of 1626 cluster galaxies located
within 105 galaxy clusters spanning a large range in redshift ($0.26 < z <
1.13)$. The galaxy clusters are massive (M$_{500} \gtrsim
2\times10^{14}$M$_{\odot}$), and are uniformly selected from the SPT and ACT
Sunyaev-Zel'dovich (SZ) surveys. With spectra in-hand for thousands of cluster
members, we use galaxies' position in projected phase space as a proxy for
their in-fall times, which provides a more robust measurement of environment
than quantities such as projected cluster-centric radius. We find clear
evidence 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 quenching
effect is found regardless of galaxy luminosity (faint or bright) and redshift
(low-$z$ or high-$z$), although the exact stellar age of galaxies depends on
both parameters at fixed environmental effects. Such a systematic increase of
$\rm D_{\rm n}4000$ with in-fall proxy would suggest that galaxies that were
accreted into hosts earlier were quenched earlier, due to longer exposure to
environmental effects such as ram pressure stripping and starvation. Compared
to the typical dynamical time scales of $1-3$ Gyr of cluster galaxies, the
relatively small age increase ($\sim$ 0.71 $\pm$ 0.4 Gyr) found in our sample
galaxies seems to suggest that a slow environmental process such as starvation
is the dominant quenching pathway. Our results provide new insights into
environmental quenching effects spanning a large range in cosmic time ($\sim
5.2$ Gyr, $z=0.26$--1.13) and demonstrate the power of using a
kinematically-derived in-fall time proxy.
