Using combined asteroseismic and spectroscopic observations of 418 red-giant stars close to the Galactic disc plane (6 kpc < RGal ≲ 13 kpc, | ZGal| < 0.3 kpc), we measure the age dependence of the radial metallicity distribution in the Milky Way's thin disc over cosmic time. The slope of the radial iron gradient of the young red-giant population (-0.058 ± 0.008 [stat.] ±0.003 [syst.] dex/kpc) is consistent with recent Cepheid measurements. For stellar populations with ages of 1-4 Gyr the gradient is slightly steeper, at a value of -0.066 ± 0.007 ± 0.002 dex/kpc, and then flattens again to reach a value of -0.03 dex/kpc for stars with ages between 6 and 10 Gyr. Our results are in good agreement with a state-of-the-art chemo-dynamical Milky-Way model in which the evolution of the abundance gradient and its scatter can be entirely explained by a non-varying negative metallicity gradient in the interstellar medium, together with stellar radial heating and migration. We also offer an explanation for why intermediate-age open clusters in the solar neighbourhood can be more metal-rich, and why their radial metallicity gradient seems to be much steeper than that of the youngest clusters. Already within 2 Gyr, radial mixing can bring metal-rich clusters from the innermost regions of the disc to Galactocentric radii of 5 to 8 kpc. We suggest that these outward-migrating clusters may be less prone to tidal disruption and therefore steepen the local intermediate-age cluster metallicity gradient. Our scenario also explains why the strong steepening of the local iron gradient with age is not seen in field stars. In the near future, asteroseismic data from the K2 mission will allow for improved statistics and a better coverage of the inner-disc regions, thereby providing tighter constraints on theevolution of the central parts of the Milky Way.