We measure the rate of environmentally-driven star formation quenching in galaxies at $zsim 1$, using eleven massive ($Mapprox 2times10^{14},mathrm{M}_odot$) 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.
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