In order to understand the roles of metal flows in galaxy formation and evolution, we analyse our self-consistent cosmological chemo-dynamical simulation of a Milky Way like galaxy during its thin-disc phase. Our simulated galaxy disc qualitatively reproduces the variation of the dichotomy in [$\alpha$/Fe]-[Fe/H] at different Galactocentric distances as derived by APOGEE-DR16, as well as the stellar age distribution in [$\alpha$/Fe]-[Fe/H] from APOKASC-2. The disc grows from the inside out, with a radial gradient in the star-formation rate during the entire phase. Despite the radial dependence, the outflow-to-infall ratio of metals in our simulated halo shows a universal (time-independent) profile scaling with the disc growth. The simulated disc undergoes two modes of gas inflow: (i) an infall of metal-poor and relatively low-[$\alpha$/Fe] gas, and (ii) a radial flow where already chemically-enriched gas moves inwards with an average velocity of $\sim0.7$ km/s. Moreover, we find that stellar migrations mostly happen outwards, on typical time scales of $\sim5$ Gyr. Our predicted radial metallicity gradients agree with the observations from APOGEE-DR16, and the main effect of stellar migrations is to flatten the radial metallicity profiles by 0.05 dex/kpc in the slopes. We also show that the effect of migrations can appear more important in [$\alpha$/Fe] than in the [Fe/H]-age relation of thin-disc stars.