  We introduce the concept of nonlocal $H$-convergence. For this we employ the
theory of abstract closed complexes of operators in Hilbert spaces. We show
uniqueness of the nonlocal $H$-limit as well as a corresponding compactness
result. Moreover, we provide a characterisation of the introduced concept,
which implies that local and nonlocal $H$-convergence coincide for
multiplication operators. We provide applications to both nonlocal and
nonperiodic fully time-dependent 3D Maxwell's equations on rough domains. The
material law for Maxwell's equations may also rapidly oscillate between eddy
current type approximations and their hyperbolic non-approximated counter
parts. Applications to models in nonlocal response theory used in quantum
theory and the description of meta-materials, to fourth order elliptic problems
as well as to homogenisation problems on Riemannian manifolds are provided.
