  We characterize general non-Markovian Gaussian maps which are covariant under
Galilean transformations. In particular, we characterize translational and
Galilean covariant maps and show that they reduce to the known Holevo result in
the Markovian limit. We apply the results to discuss measures of macroscopicity
based on classicalization maps, specifically addressing dissipation, Galilean
covariance and non-Markovianity. We further suggest a possible generalization
of the macroscopicity measure defined in Nimmrichter and Hornbergerl. [Phys.
Rev. Lett. 110, 16 (2013)].
