  We measure the projected number density profiles of galaxies and the
splashback feature in clusters selected by the Sunyaev--Zeldovich (SZ) effect
from the Advanced Atacama Cosmology Telescope (AdvACT) survey using galaxies
observed by the Dark Energy Survey (DES). The splashback radius for the
complete galaxy sample is consistent with theoretical measurements from
CDM-only simulations, and is located at $2.4^{+0.3}_{-0.4}$ Mpc $h^{-1}$. We
split the sample based on galaxy color and find significant differences in the
profile shapes. Red galaxies and those in the green valley show a
splashback-like minimum in their slope profile consistent with theoretical
predictions, while the bluest galaxies show a weak feature that appears at a
smaller radius. We develop a mapping of galaxies to subhalos in $N$-body
simulations by splitting subhalos based on infall time onto the cluster halos.
We find that the location of the steepest slope and differences in the shapes
of the profiles can be mapped to differences in the average time of infall of
galaxies of different colors. The minima of the slope in the galaxy profiles
trace a discontinuity in the phase space of dark matter halos. By relating
spatial profiles to infall time for galaxies of different colours, we can use
splashback as a clock to understand galaxy quenching. We find that red galaxies
have on average been in their clusters for over $3.2 ~\rm Gyrs$, green galaxies
about $2.2 ~\rm Gyrs$, while blue galaxies have been accreted most recently and
have not reached apocenter. Using the information from the complete radial
profiles, we fit a simple quenching model and find that the onset of galaxy
quenching in clusters occurs after a delay of about a gigayear, and that
galaxies quench rapidly thereafter with an exponential timescale of $0.6$ Gyr.
