We present a Herschel far-IR and sub-mm study of a sample of 120 galaxies in 28 Hickson Compact Groups. Fitting their UV to sub-mm spectral energy distributions with the model of da Cunha et al. (2008), we accurately estimate the dust masses, luminosities and temperatures of the individual galaxies. We find that nearly half of the late-type galaxies in dynamically old groups, those with more than 25% of early-type members and redder UV-optical colours, have also significantly lower dust-to-stellar mass ratios compared to those of actively star-forming galaxies of the same mass found both in HCGs and the field. Examining their dust-to-gas mass ratios we conclude that dust was stripped out of these systems as a result of the gravitational and hydrodynamic interactions, experienced due to previous encounters with other group members. About 40% of the early-type galaxies (mostly lenticulars), in dynamically old groups, display dust properties similar to those of the UV-optical red late-type galaxies. Given their stellar masses, star formation rates and UV-optical colours, we suggest that red late-type and dusty lenticular galaxies represent transition populations between blue star-forming disk galaxies and quiescent early-type ellipticals. [...ABRIDGED...] Our deep Herschel observations also allow us to detect the presence of diffuse cold intragroup dust in 4 HCGs. We also find that the fraction of 250micron emission which is located outside of the main bodies of the red late-type galaxies as well as of the dusty lenticulars is 15-20% of their integrated emission at this band. All these findings are consistent with an evolutionary scenario in which gas dissipation, shocks and turbulence in addition to tidal interactions, shape the evolution of galaxies in compact groups.
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