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The nanoscale periodic potentials introduced by moiré patterns in semiconducting van der Waals heterostructures have emerged as a platform for designing exciton superlattices. However, our understanding of the motion of excitons in moiré potentials is still limited. Here we investigated interlayer exciton dynamics and transport in WS 2 -WSe 2 heterobilayers in time, space and momentum domains using transient absorption microscopy combined with first-principles calculations. We found that the exciton motion is modulated by twist-angle-dependent moiré potentials around 100 meV and deviates from normal diffusion due to the interplay between the moiré potentials and strong exciton-exciton interactions. Our experimental results verified the theoretical prediction of energetically favourable K-Q interlayer excitons and showed exciton-population dynamics that are controlled by the twist-angle-dependent energy difference between the K-Q and K-K excitons. These results form a basis to investigate exciton and spin transport in van der Waals heterostructures, with implications for the design of quantum communication devices.