We investigate the role of dense Mpc-scale environments in processing molecular gas of cluster galaxies as they fall into the cluster cores . We have selected a sample of \sim 20 luminous infrared galaxies ( LIRGs ) belonging to intermediate-redshift clusters , mainly from the Hershel Lensing Survey ( HLS ) and the Local Cluster Substructure Survey ( LoCuSS ) . They include MACS J0717.5+3745 at z = 0.546 and Abell 697 , 963 , 1763 , and 2219 at z = 0.2 - 0.3 . We have performed far-infrared-to-ultraviolet spectral energy distribution modeling of the LIRGs , which span cluster-centric distances within r / r _ { 200 } \simeq 0.2 - 1.6 . We have observed the LIRGs in CO ( 1 \rightarrow 0 ) or CO ( 2 \rightarrow 1 ) with the Plateau de Bure interferometer and its successor NOEMA , as part of five observational programs carried out between 2012 and 2017 . We have compared the molecular gas to stellar mass ratio M ( H _ { 2 } ) / M _ { \star } , star formation rate ( SFR ) , and depletion time ( \tau _ { dep } ) of the LIRGs with those of a compilation of cluster and field star forming galaxies from the literature . The targeted LIRGs have SFR , M ( H _ { 2 } ) / M _ { \star } , and \tau _ { dep } that are consistent with those of both main sequence ( MS ) field galaxies and star forming galaxies from the comparison sample . However we find that the depletion time , normalized to the MS value , tentatively increases with increasing r / r _ { 200 } , with a significance of 2.8 \sigma , which is ultimately due to a deficit of cluster core LIRGs with \tau _ { dep } \gtrsim \tau _ { dep,MS } . We suggest that a rapid exhaustion of the molecular gas reservoirs occurs in the cluster LIRGs and is indeed effective in suppressing their star formation and ultimately quenching them . This mechanism may explain the exponential decrease of the fraction of cluster LIRGs with cosmic time . The compression of the gas in LIRGs , possibly induced by intra-cluster medium shocks , may be responsible for the short timescales , observed in a large fraction of cluster core LIRGs . Some of our LIRGs may also belong to a population of infalling filament galaxies .