  We quantify the star formation (SF) in the inner cores
($\mathcal{R}$/$R_{200}$$\leq$0.3) of 24 massive galaxy clusters at
0.2$\lesssim$$z$$\lesssim$0.9 observed by the $Herschel$ Lensing Survey and the
Cluster Lensing and Supernova survey with $Hubble$. These programmes, covering
the rest-frame ultraviolet to far-infrared regimes, allow us to accurately
characterize stellar mass-limited ($\mathcal{M}_{*}$$>$$10^{10}$ $M_{\odot}$)
samples of star-forming cluster members (not)-detected in the mid- and/or
far-infrared. We release the catalogues with the photometry, photometric
redshifts, and physical properties of these samples. We also quantify the SF
displayed by comparable field samples from the Cosmic Assembly Near-infrared
Deep Extragalactic Legacy Survey. We find that in intermediate-$z$ cluster
cores, the SF activity is suppressed with respect the field in terms of both
the fraction ($\mathcal{F}$) of star-forming galaxies (SFG) and the rate at
which they form stars ($\mathcal{SFR}$ and $s\mathcal{SFR} =
\mathcal{SFR}/\mathcal{M}_{*}$). On average, the $\mathcal{F}$ of SFGs is a
factor $\sim$$2$ smaller in cluster cores than in the field. Furthermore, SFGs
present average $\mathcal{SFR}$ and $s\mathcal{SFR}$ typically $\sim$0.3 dex
smaller in the clusters than in the field along the whole redshift range
probed. Our results favour long time-scale quenching physical processes as the
main driver of SF suppression in the inner cores of clusters since
$z$$\sim$0.9, with shorter time-scale processes being very likely responsible
for a fraction of the missing SFG population.
