  The Li abundance observed in pre-main sequence and main sequence late-type
stars is strongly age-dependent, but also shows a complex pattern depending on
several parameters, such as rotation, chromospheric activity and metallicity.
The best way to calibrate these effects, with the aim of studying Li as an age
indicator for FGK stars, is to calibrate coeval groups of stars, such as open
clusters (OCs) and associations. We present a considerable target sample of 42
OCs and associations, ranging from 1 Myr to 5 Gyr, observed within the Gaia-ESO
survey (GES), and using the latest data provided by GES iDR6 and the most
recent release of Gaia that was then available, EDR3. As part of this study, we
update and improve the membership analysis for all 20 OCs presented in our
previous article. We perform detailed membership analyses for all target
clusters to identify likely candidates, using all available parameters provided
by GES and based on numerous criteria: from radial velocity distributions, to
the astrometry and photometry provided by Gaia, to gravity indicators, [Fe/H]
metallicity, and Li content. We obtain updated lists of cluster members for the
whole target sample, as well as a selection of Li-rich giant contaminants
obtained as an additional result of the membership process. Each selection of
cluster candidates was thoroughly contrasted with numerous existing membership
studies using data from Gaia to ensure the most robust results. These final
cluster selections will be used in the third and last paper of this series,
which reports the results of a comparative study characterising the observable
Li dispersion in each cluster and analysing its dependence on several
parameters, allowing us to calibrate a Li-age relation and obtain a series of
empirical Li envelopes for key ages in our sample.
