We investigate the role of dense megaparsec-scale environments in processing molecular gas of cluster galaxies as they fall into the cluster cores. We selected a sample of similar to 20 luminous infrared galaxies (LIRGs) belonging to intermediate-redshift clusters, mainly from the Herschel Lensing Survey and the Local Cluster Substructure Survey. These galaxies include MACS J0717.5+3745 at z=0.546 and Abell 697, 963, 1763, and 2219 at z=0.2-0.3. We performed spectral energy distribution modeling from the far-infrared to ultraviolet of the LIRGs, which span cluster-centric distances within r/r(200)similar or equal to 0.2-1.6. We observed the LIRGs in CO(1 -> 0) or CO(2 -> 1) with the Plateau de Bure interferometer and its successor NOEMA, as part of five observational programs carried out between 2012 and 2017. We 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)greater than or similar to 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 that are observed in a large fraction of cluster-core LIRGs. Some of our LIRGs may also belong to a population of infalling filament galaxies.