This paper focuses on carbon, which is one of the most abundant elements in the universe and is of high importance in the field of nucleosynthesis and galactic and stellar evolution. The origin of carbon and the relative importance of massive and low- to intermediate-mass stars in producing it is still a matter of debate. We aim at better understanding the origin of carbon by studying the trends of [C/H], [C/Fe], and [C/Mg] versus [Fe/H] and [Mg/H] for 2133 FGK dwarf stars from the fifth Gaia-ESO Survey internal data release (GES iDR5). The availability of accurate parallaxes and proper motions from Gaia DR2 and radial velocities from GES iDR5 allows us to compute Galactic velocities, orbits, absolute magnitudes, and, for 1751 stars, Bayesian-derived ages. Three different selection methodologies have been adopted to discriminate between thin- and thick-disk stars. In all the cases, the two stellar groups show different [C/H], [C/Fe], and [C/Mg] and span different age intervals, with the thick-disk stars being, on average, older than the thin-disk ones. The behaviors of [C/H], [C/Fe], and [C/Mg] versus [Fe/H], [Mg/H], and age all suggest that C is primarily produced in massive stars. The increase of [C/Mg] for young thin-disk stars indicates a contribution from low-mass stars or the increased C production from massive stars at high metallicities due to the enhanced mass loss. The analysis of the orbital parameters R-med and supports an "inside-out" and "upside-down" formation scenario for the disks of the Milky Way.