We report in situ Raman scattering experiments on single-layer graphene (SLG) and Bernal bilayer graphene (BLG) during exposure to rubidium vapor. The G- and 2D-band evolutions with doping time are presented and analyzed. On SLG, the extended doping range scanned (up to about 1014 electrons/cm2) allows the observation of three regimes in the evolution of the G-band frequency: a continuous upshift followed by a plateau and a downshift. Overall the measured evolution is interpreted as the signature of the competition between dynamic and adiabatic effects upon n-doping. Comparison of the obtained results with theoretical predictions indicates however that a substrate pinning effect occurs and inhibits charge-induced lattice expansion of SLG. At low doping, a direct link between electrostatic gating and Rb doping results is presented. For BLG, the added electrons are shown to be first confined in the top layer, but the system evolves with time toward a more symmetric repartition of the added electrons in both layers. The results obtained on BLG also confirm that the slope of the phonon dispersion close to the K point tends to be slightly reduced at low doping but suggest the occurrence of an unexpected increase of the phonon dispersion slope at higher electron concentration.