  Motivated by the recent experiments and the wide tunability on the
MoTe$_2$/WSe$_2$ moir\'{e} heterobilayer, we consider a physical model to
explore the underlying physics for the interplay between the itinerant carriers
and the local magnetic moments. In the regime where the MoTe$_2$ is tuned to a
triangular lattice Mott insulator and the WSe$_2$ layer is doped with the
itinerant holes, we invoke the itinerant ferromagnetism from the double
exchange mechanism for the itinerant holes on the WSe$_2$ layer and the local
moments on the MoTe$_2$ layer. Together with the antiferromagnetic exchange on
the MoTe$_2$ layer, the itinerant ferromagnetism generates the scalar spin
chirality distribution in the system. We further point out the presence of
spin-assisted hopping in addition to the Kondo coupling between the local spin
and the itinerant holes, and demonstrate the topological Hall effect for the
itinerant electrons in the presence of the non-collinear spin configurations.
This work may improve our understanding of the correlated moir\'{e} systems and
inspire further experimental efforts.
