  The theoretical description of the external degrees of freedom of atoms
trapped inside a magneto-optical trap (MOT) often relies on the decoupling of
the evolution of the internal and external degrees of freedom. That is possible
thanks to much shorter timescales typically associated with the first ones. The
electronic structure of alkaline-earth atoms, on the other hand, presents
ultra-narrow transitions and metastable states that makes such an approximation
invalid in the general case. In this article, we report on a model based on
open Bloch equations for the evolution of the number of atoms in a
magneto-optical trap. With this model we investigate the loading of the
strontium blue magneto-optical trap under different repumping schemes, either
directly from a Zeeman slower, or from an atomic reservoir made of atoms in a
metastable state trapped in the magnetic quadrupolar field. The fluorescence
observed on the strong 461~nm transition is recorded and quantitatively
compared with the results from our simulations. The comparison between
experimental results and calculations within our model allowed to identify the
existence of the decay paths between the upper level of the repumping
transition and the dark strontium metastable states, which could not be
explained by electric dipole transition rates calculated in the literature.
Moreover, our analysis pinpoints the role of the atomic movement in limiting
the efficiency of the atomic repumping of the Sr metastable states.
