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We use equivariant localization and divided difference operators to determineformulas for the torus-equivariant fundamental cohomology classes of $K$-orbitclosures 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 varietywhich is equipped with a single flag of subbundles and which splits as a directsum of subbundles of ranks $p$ and $q$. The precise description of such adegeneracy locus relies upon knowing a set-theoretic description of $K$-orbitclosures, which we provide via a detailed combinatorial analysis of the posetof "$(p,q)$-clans," which parametrize the orbit closures. We describe preciselyhow our formulas for the equivariant classes of $K$-orbit closures can beinterpreted as formulas for the classes of such degeneracy loci in the Chernclasses of the bundles involved. In the cases outside of type $A$, we suggestthat the orbit closures should parametrize degeneracy loci involving a vectorbundle equipped with a non-degenerate symmetric or skew-symmetric bilinearform, a single flag of subbundles which are isotropic or Lagrangian withrespect to the form, and a splitting as a direct sum of subbundles with eachsummand satisfying some property (depending on $K$) with respect to the form.The precise description of such a degeneracy locus is conjectured for all casesin types $B$ and $C$.

K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles splitting as direct sums