  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 $2.8\sigma$, which is ultimately
due to a deficit of cluster core LIRGs with $\tau_{\rm dep}\gtrsim\tau_{\rm
dep,MS}$. 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.
