  Let $H\subset G$ be semisimple Lie groups, $\Gamma\subset G$ a lattice and
$K$ a compact subgroup of $G$. For $n \in \mathbb N$, let $\mathcal O_n$ be the
projection to $\Gamma \backslash G/K$ of a finite union of closed $H$-orbits in
$\Gamma \backslash G$. In this very general context of homogeneous dynamics, we
prove an equidistribution theorem for intersections of $\mathcal O_n$ with an
analytic subvariety $S$ of $G/K$ of complementary dimension: if $\mathcal O_n$
is equidistributed in $\Gamma \backslash G/K$, then the signed intersection
measure of $S \cap \mathcal O_n$ normalized by the volume of $\mathcal O_n$
converges to the restriction to $S$ of some $G$-invariant closed form on $G/K$.
We give general tools to determine this closed form and compute it in some
examples.
  As our main application, we prove that, if $\mathbb V$ is a polarized
variation of Hodge structure of weight $2$ and Hodge numbers $(q,p,q)$ over a
base $S$ of dimension $rq$, then the (non-exceptional) 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 $q^{\textrm{th}}$ Chern form of the Hodge bundle.
This generalizes a previous work of the first author which treated the case
$q=r=1$. We also prove an equidistribution theorem for certain families of CM
points in Shimura varieties, and another one for Hecke translates of a divisor
in $\mathcal A_g$.
