The recent observation (Gorbachev et al 2014 Science 346 448) of nonlocal resistance RNL near the Dirac point (DP) of multiterminal graphene on aligned hexagonal-boron nitride (G/hBN) has been interpreted as the consequence of topological valley Hall currents carried by the Fermi sea states just beneath the bulk gap Eg induced by inversion symmetry breaking. However, the corresponding valley Hall conductivity , quantized inside Eg, is not directly measurable. Conversely, the Landauer–Büttiker formula, as a numerically exact approach to observable nonlocal transport quantities, yields RNL ≡ 0 for the same simplistic Hamiltonian of gapped graphene that generates via the Kubo formula. We combine ab initio with quantum transport calculations to demonstrate that G/hBN wires with zigzag edges host dispersive edge states near the DP that are absent in theories based on the simplistic Hamiltonian. Although such edge states exist also in isolated zigzag graphene wires, aligned hBN is required to modify their energy–momentum dispersion and generate near the DP. The Fermi surface-determined edge currents carrying the nonlocal signal persist also in the presence of edge disorder and over long distances. Concurrently, they resolve the long-standing puzzle of why the highly insulating state of G/hBN is rarely observed. Thus, we conclude that the observed RNL is unrelated to Fermi sea topological valley currents conjectured for gapped Dirac spectra.