  We measure FIR emission from tails of stripped dust following the ionised and
atomic gas components in galaxies undergoing ram pressure stripping. We study
the dust-to-gas relative distribution and mass ratio in the stripped
interstellar medium and relate them to those of the intra-cluster medium, thus
linking the cluster-ICM-galaxy evolution at small-scales. The galaxy sample
consists of three Scd Virgo galaxies with stellar masses in the range
$10^9\lesssim \mathrm{M_{*}} \lesssim 10^{10}\, \mathrm{M_{\odot}}$, and within
1 Mpc from the cluster centre, namely NGC 4330, NGC 4522, and NGC 4654. Through
the analysis of VESTIGE H$\alpha$, $Herschel$ SPIRE far-infrared, and VIVA HI
data, we trace the spatial distribution of the tails and infer the dust and gas
masses from the measured far-infrared 250 $\mu$m and HI flux densities.
Dust-to-gas mass ratios (DGRs) in the tails are analysed as a function of the
galaxy mass, metallicity, and dust temperature. Along the stripped component,
the dust distribution closely follows the HI and H$\alpha$ emitting gas, all
extending beyond the optical disc. In these regions, the DGRs are
$2.0\pm0.6\times10^{-3}$, $0.7\pm0.1\times10^{-3}$, and
$0.4\pm0.03\times10^{-3}$, for NGC 4330, NGC 4522, and NGC 4654, respectively,
i.e. up to a factor of 15 less than the values measured in the main body of
nearby galaxies. We also find a negative trend in the DGR as a function of the
metallicity that can be explained in terms of a dust component more centrally
concentrated in more metal-rich systems. Together with the finding that the
stripped dust is cold, $T_{d} \lesssim 25\, K$, our results support an
outside-in stripping scenario of the galaxy interstellar medium. This study
shows that ram pressure stripping is a key mechanism in the building up of the
Virgo intra-cluster component injecting dust grains into the ICM, thus
contributing to its metal enrichment.
