  We use equivariant localization and divided difference operators to determine
formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit
closures on the flag variety $G/B$ for various symmetric pairs $(G,K)$. In type
$A$, we realize the closures of $K=GL(p,\C) \times GL(q,\C)$-orbits on
$GL(p+q,\C)/B$ as universal degeneracy loci for a vector bundle over a variety
which is equipped with a single flag of subbundles and which splits as a direct
sum of subbundles of ranks $p$ and $q$. The precise description of such a
degeneracy locus relies upon knowing a set-theoretic description of $K$-orbit
closures, which we provide via a detailed combinatorial analysis of the poset
of "$(p,q)$-clans," which parametrize the orbit closures. We describe precisely
how our formulas for the equivariant classes of $K$-orbit closures can be
interpreted as formulas for the classes of such degeneracy loci in the Chern
classes of the bundles involved. In the cases outside of type $A$, we suggest
that the orbit closures should parametrize degeneracy loci involving a vector
bundle equipped with a non-degenerate symmetric or skew-symmetric bilinear
form, a single flag of subbundles which are isotropic or Lagrangian with
respect to the form, and a splitting as a direct sum of subbundles with each
summand satisfying some property (depending on $K$) with respect to the form.
The precise description of such a degeneracy locus is conjectured for all cases
in types $B$ and $C$.
