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 $10^9lesssim mathrm{M_{*}} lesssim 10^{10}, mathrm{M_{odot}}$, 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 $2.0pm0.6times10^{-3}$, $0.7pm0.1times10^{-3}$, and $0.4pm0.03times10^{-3}$, 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.
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