We investigate the role of dense Mpc-scale environments in processing molecular gas of cluster galaxies as they fall into the cluster cores. We consider $sim20$ luminous infrared galaxies (LIRGs) in intermediate-$z$ clusters, from the Hershel Lensing Survey and the Local Cluster Substructure Survey. 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 $r/r_{200}simeq0.2-1.6$. 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 ($tau_{rm dep}$) of the LIRGs with those of a compilation of cluster and field star forming galaxies. The targeted LIRGs have SFR, $M(H_2)/M_star$, and $tau_{rm dep}$ 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, increases with increasing $r/r_{200}$, with a significance of $2.8sigma$, which is ultimately due to a deficit of cluster core LIRGs with $tau_{rm dep}gtrsimtau_{rm dep,MS}$. We suggest that a rapid exhaustion of the molecular gas reservoirs occurs in the cluster LIRGs and is effective in suppressing their star formation. 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 depletion timescales, observed in a large fraction of cluster core LIRGs.
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