  We present self-consistent calculations coupling numerical hydrodynamics
simulations of collapsing pre-stellar cores and stellar evolution models of
accreting objects. We analyse the main impact of consistent accretion history
on the evolution and lithium depletion of young low-mass stars and brown
dwarfs. These consistent models confirm the generation of a luminosity spread
in Herzsprung-Russell diagrams at ages $\sim$ 1-10 Myr. They also confirm that
early accretion can produce objects with abnormal Li depletion, as found in a
previous study that was based on arbitrary accretion rates. The results
strengthen that objects with anomalously high level of Li depletion in young
clusters should be extremely rare. We also find that early phases of burst
accretion can produce coeval models of similar mass with a range of different
Li surface abundances, and in particular with Li-excess compared to the
predictions of non-accreting counterparts. This result is due to a subtle
competition between the effect of burst accretion and its impact on the central
stellar temperature, the growth of the stellar radiative core and the accretion
of fresh Li from the accretion disk. Only consistent models could reveal such a
subtle combination of effects. This new result could explain the recent,
puzzling observations of Li-excess of fast rotators in the young cluster NGC
2264. Present self-consistent accreting models are available in electronic
form.
