  Let $G/H$ be a homogeneous variety, and let $X$ be a $G$-equivariant
embedding of $G/H$such that the number of $G$-orbits in $X$ is finite. We show
that the equivariant Borel-Moore homology of $X$ has a filtration with
associated graded module the direct sum of the equivariant Borel-Moore
homologies of the $G$-orbits. If $T$ is a maximal torus of $G$ such that each
$G$-orbit has a $T$-fixed point, then the equivariant filtration descends to
give a filtration on the ordinary Borel-Moore homology of $X$. We apply our
findings to certain wonderful compactifications as well as to double flag
varieties.
