  The act of measuring a system has profound consequences of dynamical and
thermodynamic nature. In particular, the degree of irreversibility ensuing from
a non-equilibrium process is strongly affected by measurements aimed at
acquiring information on the state of a system of interest: the conditional and
unconditional entropy production, which quantify the degree of irreversibility
of the open system's dynamics, are related to each other by clearly interpreted
informational quantities. Building on a recently proposed collisional-model
framework [G. T. Landi {\it et al.}, arXiv:2103.06247], we investigate the case
of continuous-variable information carriers prepared in Gaussian states and
undergoing Gaussian processes. We build up a toolbox that fully characterizes
the thermodynamics of continuously measured non-equilibrium Gaussian systems
and processes, illustrating how the instruments hereby introduced provide key
insight into recent experiments on mesoscopic quantum systems [Phys. Rev. Lett,
{\bf 125}, 080601 (2020)].
