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, and 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 - with an age range from 1Myr to 5Gyr - 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, complemented with detailed bibliographical searches, and based on numerous criteria: from radial velocity distributions, to the astrometry (proper motions and parallaxes) and photometry provided by Gaia, to gravity indicators (logg and the gamma index), [Fe/H] metallicity, and Li content in diagrams of EW(Li) (Li equivalent widths) versus Teff. 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.