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We use first-principle density functional theory (DFT) to study the transportproperties of single and double barrier heterostructures realized by stackingmultilayer h-BN or BC$_{2}$N, and graphene films between graphite leads. Theheterostructures are lattice matched. The considered single barrier systemsconsist of layers of up to five h-BN or BC$_{2}$N monoatomic layers (Bernalstacking) between graphite electrodes. The transmission probability of an h-BNbarrier exhibits two unusual behaviors: it is very low also in a classicallyallowed energy region, due to a crystal momentum mismatch between states ingraphite and in the dielectric layer, and it is only weakly dependent on energyin the h-BN gap, because the imaginary part of the crystal momentum of h-BN isalmost independent of energy. The double barrier structures consist of h-BNfilms separated by up to three graphene layers. We show that already fivelayers of h-BN strongly suppress the transmission between graphite leads, andthat resonant tunneling cannot be observed because the energy dispersionrelation cannot be decoupled in a vertical and a transversal component.

Tunneling properties of vertical heterostructures of multilayer hexagonal boron nitride and graphene