Asteroseismology provides a unique opportunity to probe the interiors of evolved stars and constrain their internal rotation. The correct reproduction of the core rotation evolution is key to understanding the internal processes involved in low-mass stars. We explore the efficiency required to reproduce the behaviour of the transport of angular momentum (AM) in view of asteroseismic constraints. We computed a series of models and investigated an updated AM transport by including a time-dependent extra viscosity related to the AMRI. We compared our predictions to the asteroseismic measurements of the core and surface rotation of a sample of SGB and RGB stars. We confirm that a time-dependent additional viscosity is required to reproduce the general behaviour of the core rotation rate along evolution. We show that it results in stronger Li and Be depletions for low-mass stars over evolution. We confirm that predicted Li abundances at the RGB bump by classical models, commonly used as references, cannot reproduce the Li depletion along the MS and evolved phases of stellar evolution. We show that the observed amount of Li of stars less massive than 1Msun leads to a discrepancy between model predictions and observations at the RGB bump. We show that a semi-parametric model can reproduce the rotational behaviour along the first phases of evolution well, with the exception of the sharp transition observed during the SGB phase. This suggests that two distinct transport processes may be involved. The processes required to transport chemicals during the MS, and AM until the RGB phase impact the Li depletion all along the evolutionary duration. A good prediction of the Li abundance at young phases places strong constraints on the predicted one at more evolved phases. It also impacts the threshold that defines Li-rich giant stars, showing that classical models tend to overestimate its value.