  Using combined asteroseismic and spectroscopic observations of 418 red-giant
stars close to the Galactic disc plane (6 kpc $<R_{\rm Gal}\lesssim13$ kpc,
$|Z_{\rm Gal}|<0.3$ 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
($-0.058\pm0.008$ [stat.] $\pm0.003$ [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 $-0.066\pm0.007\pm0.002$ dex/kpc,
and then flattens again to reach a value of $\sim-0.03$ 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 the evolution of the central parts of
the Milky Way.
