We present an analysis of Li abundances in low mass stars (LMS) during the Red Giant Branch (RGB) and Asymptotic Giant Branch (AGB) stages, based on a new determination of their luminosities and evolutionary status. By applying recently suggested models for extra-mixing, induced by magnetic buoyancy, we show that both Li-rich and Li-poor stars can be accounted for. The simplest scenario implies the development of fast instabilities on the RGB, where Li is produced. When the fields increase in strength, buoyancy slows down and Li is destroyed. ³He is consumed, at variable rates. The process continues on the AGB, where however moderate mass circulation rates have little effect on Li due to the short time available. O-rich and C-rich stars show different histories of Li production/destruction, possibly indicative of different masses. More complex transport schemes are allowed by magnetic buoyancy, with larger effects on Li, but most normal LMS seem to show only the range of Li variation discussed here.