  Coulomb coupling between proximal layers in graphene heterostructures results
in efficient energy transfer between the layers. We predict that, in the
presence of correlated density inhomogeneities in the layers, vertical energy
transfer has a strong impact on lateral charge transport. In particular, for
Coulomb drag it dominates over the conventional momentum drag near zero doping.
The dependence on doping and temperature, which is different for the two drag
mechanisms, can be used to separate these mechanisms in experiment. We predict
distinct features such as a peak at zero doping and a multiple sign reversal,
which provide diagnostics for this new drag mechanism.
