  We test a scenario in which radial migration could affect the Li abundance
pattern of dwarf stars in the solar neighbourhood. This may confirm that the Li
abundance in these stars can not serve as a probe for the Li abundance in the
interstellar medium. We use the high-quality data (including Li abundances)
from the 6th internal Data Release of the Gaia-ESO survey. In this sample, we
group stars by similarity in chemical abundances via hierarchical clustering.
Our analysis treats both measured Li abundances and upper limits. The Li
envelope of the previously identified radially migrated stars is well below the
benchmark meteoritic value (<3.26 dex); the star with the highest detected
abundance has A(Li) = 2.76 dex. This confirms the previous trends observed for
old dwarf stars (median ages $\sim$ 8 Gyr), where Li decreases for
[Fe/H]$\gtrsim$0. This result acts as supporting evidence that the abundance of
Li measured in the upper envelope of old dwarf stars should not be considered a
proxy for the interstellar medium Li. Our scenario also indicates that the
stellar yields for [M/H]>0 should not be decreased, as recently proposed in the
literature. Our study backs the recent studies that claimed that old dwarfs on
the hot side of the dip are efficient probes of the ISM abundance of Li,
provided atomic diffusion does not lower significantly the initial Li abundance
in the atmospheres of metal-rich objects.
