  We study branching mappings that satisfy some condition of distortion of the
modulus of families of paths. In a situation where the definition domain of
mappings is locally connected on its boundary, the mapped domain is regular,
and the majorant responsible for distortion of the modulus of families of paths
is integrable, it is proved that the families of all specified mappings with
one normalization condition are equicontinuous in the closure of the given
domain.
