  Lithium is extensively known to be a good tracer of non-standard mixing
processes occurring in stellar interiors. We present the results of a new large
Lithium survey in red giant stars and combine it with surveys from the
literature to probe the impact of rotation-induced mixing and thermohaline
double-diffusive instability along stellar evolution. We determined the surface
Li abundance for a sample of 829 giant stars with accurate Gaia parallaxes for
a large sub-sample (810 stars) complemented with accurate Hipparcos parallaxes
(19 stars). The spectra of our sample of northern and southern giant stars were
obtained in three ground-based observatories (OHP, ESO-La Silla, Mc Donald). We
determined the atmospheric parameters (Teff, log(g), [Fe/H]), and the Li
abundance. We used Gaia parallaxes and photometry to determine the luminosity
of our objects and we estimated the mass and evolution status of each sample
star with a maximum-likelihood technique using stellar evolution models
computed with the STAREVOL code. We compared the observed Li behaviour with
predictions from stellar models, including rotation and thermohaline mixing.
The same approach was used for stars from selected Li surveys from the
literature. Rotation-induced mixing accounts nicely for the lithium behaviour
in stars warmer than about 4200K, independently of the mass domain. For stars
with masses lower than 2Msun thermohaline mixing leads to further Li depletion
below the Teff of the RGB bump (about 4000K), and on the early AGB, as
observed. Depending on the definition we adopt, we find between 0.8 and 2.2% of
Li-rich giants in our new sample. Gaia puts a new spin on the understanding of
mixing processes in stars, and our study confirms the importance of
rotation-induced processes and of thermohaline mixing. However asteroseismology
is required to definitively pinpoint the actual evolution status of Li-rich
giants.
