  All local electronic properties of graphene on a hexagonal boron nitride
(hBN) substrate exhibit spatial moir\'e patterns related to lattice constant
and orientation differences between shared triangular Bravais lattices. We
apply a previously derived effective Hamiltonian for the $\pi$-bands of
graphene on h-BN to address the carrier-dependence of transport properties,
concentrating on the conductivity features at four electrons and four holes per
unit cell. These transport features measure the strength of Bragg scattering of
$\pi$-electrons off the moir\'e pattern, and exhibit a striking particle-hole
asymmetry that we trace to specific features of the effective Hamiltonian that
we interpret physically.
