  We measure the rate of environmentally-driven star formation quenching in
galaxies at $z\sim 1$, using eleven massive ($M\approx
2\times10^{14}\,\mathrm{M}_\odot$) galaxy clusters spanning a redshift range
$1.0<z<1.4$ from the GOGREEN sample. We identify three different types of
transition galaxies: "green valley" (GV) galaxies identified from their
rest-frame $(NUV-V)$ and $(V-J)$ colours; "blue quiescent" (BQ) galaxies, found
at the blue end of the quiescent sequence in $(U-V)$ and $(V-J)$ colour; and
spectroscopic post-starburst (PSB) galaxies. We measure the abundance of these
galaxies as a function of stellar mass and environment. For high stellar mass
galaxies ($\log{M/\mathrm{M}_\odot}>10.5$) we do not find any significant
excess of transition galaxies in clusters, relative to a comparison field
sample at the same redshift. It is likely that such galaxies were quenched
prior to their accretion in the cluster, in group, filament or protocluster
environments. For lower stellar mass galaxies
($9.5<\log{M/\mathrm{M}_\odot}<10.5$) there is a small but significant excess
of transition galaxies in clusters, accounting for an additional $\sim 5-10$
per cent of the population compared with the field. We show that our data are
consistent with a scenario in which 20--30 per cent of low-mass, star-forming
galaxies in clusters are environmentally quenched every Gyr, and that this rate
slowly declines from $z=1$ to $z=0$. While environmental quenching of these
galaxies may include a long delay time during which star formation declines
slowly, in most cases this must end with a rapid ($\tau<1$ Gyr) decline in star
formation rate.
