  Let $\mathbb V$ be a polarized variation of Hodge structure over a smooth
complex quasi-projective variety $S$. In this paper, we give a complete
description of the typical Hodge locus for such variations. We prove that it is
either empty or equidistributed with respect to a natural differential form,
\emph{the pull-push form}. In particular, it is always analytically dense when
the pull-push form does not vanish. When the weight is $2$, the Hodge numbers
are $(q,p,q)$ and the dimension of $S$ is least $rq$, we prove that the typical
locus where the Picard rank is at least $r$ is equidistributed in $S$ with
respect to the volume form $c_q^r$, where $c_q$ is the $q$\textsuperscript{th}
Chern form of the Hodge bundle. We obtain also several equidistribution results
of the typical locus in Shimura varieties: a criterion for the density of the
typical Hodge loci of a variety in $\mathcal{A}_g$, equidistribution of certain
families of CM points and equidistribution of Hecke translates of curves and
surfaces in $\mathcal A_g$.
  These results are proved in the much broader context of dynamics on
homogeneous spaces of Lie groups which are of independent interest. The
pull-push form appear in this greater generality and we provide several tools
to determine it and we compute it in many examples.
