  We performed a detailed study of the evolution of the luminosity of
He-ignition stage and of the red giant branch bump luminosity during the red
giant branch phase transition for various metallicities. To this purpose we
calculated a grid of stellar models that sample the mass range of the
transition with a fine mass step equal to ${\rm 0.01M_\odot}$. We find that for
a stellar population with a given initial chemical composition, there is a
critical age (of 1.1-1.2~Gyr) around which a decrease in age of just 20-30
million years causes a drastic drop in the red giant branch tip brightness. We
also find a narrow age range (a few $10^7$ yr) around the transition,
characterized by the luminosity of the red giant branch bump being brighter
than the luminosity of He ignition. We discuss a possible link between this
occurrence and observations of Li-rich core He-burning stars.
