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 0.01 M⊙. 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 107 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.