  We investigate the connection between environment and the different quenching
channels that galaxies are prone to follow in the rest-frame NUVrK colour
diagram, as identified by Moutard et al. (2016b). Namely, the fast quenching
channel followed by $young$ low-mass galaxies and the slow quenching channel
followed by $old$ high-mass ones. We make use of the >22 deg$^2$ covered the
VIPERS Multi-Lambda Survey (VIPERS-MLS) to select a galaxy sample complete down
to stellar masses of $M_* > 10^{9.4} M_\odot$ at $z < 0.65$ ($M_* > 10^{8.8}
M_\odot$ at $z < 0.5$) and including 33,500 (43,000) quiescent galaxies
properly selected at $0.2 < z < 0.65$, while being characterized by reliable
photometric redshifts ($\sigma_{\delta z/(1+z)} \leq 0.04$) that we use to
measure galaxy local densities. We find that (1) the quiescence of low-mass
[$M_* \leq 10^{9.7} M_\odot$] galaxies requires a strong increase of the local
density, which confirms the lead role played by environment in their fast
quenching and, therefore, confirms that the low-mass upturn observed in the
stellar mass function of quiescent galaxies is due to $environmental$
$quenching$. We also observe that (2) the reservoir of low-mass star-forming
galaxies located in very dense regions (prone to environmental quenching) has
grown between $z \sim 0.6$ and $ z \sim 0.4$ whilst the share of low-mass
quiescent galaxies (expected to being environmentally quenched) may have
simultaneously increased, which would plead for a rising importance of
$environmental$ $quenching$ with cosmic time, compared to $mass$ $quenching$.
We finally discuss the composite picture of such environmental quenching of
low-mass galaxies and, in particular, how this picture may be consistent with a
$delayed$-$then$-$rapid$ quenching scenario.
