  Lithium-rich giant stars are rare and their existence challenges our
understanding of stellar structure and evolution. We profit from the
high-quality sample gathered with HARPS and UVES, in order to search for
Li-rich giants and to identify the Li enrichment mechanisms responsible. We
derive stellar parameters for 247 stars belonging to 32 open clusters, with
0.07 Ga < ages < 3.6 Ga. We employed the spectral synthesis technique code
FASMA for the abundance analysis of 228 stars from our sample. We also
determined ages, distances, and extinction using astrometry and photometry from
Gaia and PARSEC isochrones to constrain their evolutionary stage. Our sample
covers a wide range of stellar masses from 1 to more than 6 solar masses where
the majority of the masses are above 2 solar masses. We have found 14 canonical
Li-rich giant stars which have experienced the first dredge-up. This
corresponds to 6% of our total sample, which is higher than what is typically
found for field stars. Apart from the canonical limit, we use the maximum Li
abundance of the progenitor stars as a criterion for Li enrichment. We find Li
enhancement also among eight stars which have passed the first dredge up and
show strong Li lines based on the fact that stars at the same evolutionary
stage in the same cluster have significantly different Li abundances. We
confirm that giants with higher Li abundance correspond to a higher fraction of
fast-rotating giants, suggesting a connection between Li enhancement and
stellar rotation as predicted by stellar models. Our Li-rich giants are found
in various evolutionary stages implying that no unique Li production mechanism
is responsible for Li enrichment but rather different intrinsic or external
mechanisms can be simultaneously at play.
