  We aim to constrain the mixing processes in low-mass stars by investigating
the behaviour of the Li surface abundance after the main sequence. We take
advantage of the data from the sixth internal data release of Gaia-ESO, idr6,
and from the Gaia Early Data Release 3, edr3. We select a sample of main
sequence, sub-giant, and giant stars in which Li abundance is measured by the
Gaia-ESO survey, belonging to 57 open clusters with ages from 120~Myr to about
7 Gyr and to Milky Way fields, covering a range in [Fe/H] between -1.0 and
+0.5dex. We study the behaviour of the Li abundances as a function of stellar
parameters. We compare the observed Li behaviour in field giant stars and in
giant stars belonging to individual clusters with the predictions of a set of
classical models and of models with mixing induced by rotation and thermohaline
instability. The comparison with stellar evolution models confirms that
classical models cannot reproduce the lithium abundances observed in the
metallicity and mass regimes covered by the data. The models that include the
effects of both rotation-induced mixing and thermohaline instability account
for the Li abundance trends observed in our sample, in all metallicity and mass
ranges. The differences between the results of the classical models and of the
rotation models largely differ (up to ~2 dex), making lithium the best element
to constrain stellar mixing processes in low-mass stars. For stars with
well-determined masses, we find a better agreement between observed surface
abundances and models with rotation-induced and thermohaline mixings, the
former dominating during the main sequence and the first phases of the
post-main sequence evolution and the latter after the bump in the luminosity
function.
