  We use the semi-analytic model of galaxy formation SAG to study the relevance
of mass and environmental quenching on satellite galaxies. We find that
environmental processes dominate the star formation (SF) quenching of low-mass
satellites ($M_{\star} \lesssim 10^{10.5}\, {\rm M}_{\odot}$), whereas
high-mass galaxies typically quench as centrals. High-mass galaxies that remain
actively forming stars while being accreted are found to be mainly affected by
mass quenching after their first infall. For a given stellar mass, our model
predicts SF quenching to be less efficient in low-mass haloes both before and
after infall, in contradiction with common interpretations of observational
data. Our model supports a two-stage scenario to explain the SF quenching.
Initially, the SF of satellites resembles that of centrals until the gas
cooling rate is reduced to approximately half its value at infall. Then, the SF
fades through secular processes that exhaust the cold gas reservoir. This
reservoir is not replenished efficiently due to the action of either
ram-pressure stripping (RPS) of the hot gas in low-mass satellites, or feedback
from the active galactic nucleus (AGN) in high-mass satellites. The delay times
for the onset of SF quenching are found to range from $\approx 3\,{\rm Gyr}$ to
$\approx 1\,{\rm Gyr}$ for low-mass ($M_{\star} \approx 10^{10}\, {\rm
M}_{\odot}$) and high-mass ($M_{\star} \approx 10^{11}\, {\rm M}_{\odot}$)
satellites, respectively. SF fades in $\approx 1.5\,{\rm Gyr}$, largely
independent of stellar mass. We find that the SF quenching of low-mass
satellites supports the so-called delay-then-rapid quenching scenario. However,
the SF history of $z=0$ passive satellites of any stellar mass is better
described by a delay-then-fade quenching scenario.
