The presence of multiple populations in globular clusters has been well established thanks to high-resolution spectroscopy. It is widely accepted that distinct populations are a consequence of different stellar generations: intracluster pollution episodes are required to produce the peculiar chemistry observed in almost all clusters. Unfortunately, the progenitors responsible have left an ambiguous signature and their nature remains unresolved. To constrain the candidate polluters, we have measured lithium and aluminium abundances in more than 180 giants across three systems: NGC 1904, NGC 2808, and NGC 362. The present investigation along with our previous analysis of M12 and M5 affords us the largest data base of simultaneous determinations of Li and Al abundances. Our results indicate that Li production has occurred in each of the three clusters. In NGC 362, we detected an M12-like behaviour, with first- and second-generation stars sharing very similar Li abundances favouring a progenitor that is able to produce Li, such as asymptotic giant branches stars. Multiple progenitor types are possible in NGC 1904 and NGC 2808, as they possess both an intermediate population comparable in lithium to the first generation stars and also an extreme population, that is enriched in Al but depleted in Li. A simple dilution model fails in reproducing this complex pattern. Finally, the internal Li variation seems to suggest that the production efficiency of this element is a function of the cluster's mass and metallicity - low-mass or relatively metal-rich clusters are more adept at producing Li.