$H$-compactness for nonlocal linear operators in fractional divergence form

We study the $H$-convergence of nonlocal linear operators in fractional divergence form, where the oscillations of the matrices are prescribed outside the reference domain. Our compactness argument bypasses the failure of the classical localisation techniques that mismatch with the nonlocal nature of the operators involved. If symmetry is also assumed, we extend the equivalence between the $H$-convergence of the operators and the $\mathrm{\Gamma }$-convergence of the associated energies.