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 up to z ∼ 0.65 (M * > 10 8.8 M up to 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 (σ δz/(1+z) ≤ 0.04) that we use to measure galaxy local densities. We find that (1) the quiescence of low-mass [M * ≤ 10 9.7 M ] 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 ∼ 0.6 and z ∼ 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.