  Van der Waals heterostructures constitute a platform for investigating
intriguing many-body quantum phenomena. In particular, transition-metal
dichalcogenide (TMD) hetero-bilayers host long-lived interlayer excitons which
exhibit permanent out-of-plane dipole moments. Here, we develop a microscopic
theory for interlayer exciton-exciton interactions including both the dipolar
nature of interlayer excitons as well as their fermionic substructure, which
gives rise to an attractive fermionic exchange. We find that these interactions
contribute to a drift force resulting in highly non-linear exciton propagation
at elevated densities in the MoSe$_2$-WSe$_2$ heterostructure. We show that the
propagation can be tuned by changing the number of hBN spacers between the TMD
layers or by adjusting the dielectric environment. In particular, although
counter-intuitive, we reveal that interlayer excitons in free-standing samples
propagate slower than excitons in hBN-encapsulated TMDs - due to an enhancement
of the net Coulomb-drift with stronger environmental screening. Overall, our
work contributes to a better microscopic understanding of the interlayer
exciton transport in technologically promising atomically thin semiconductors.
