Interlayer exciton dynamics in a van der Waals heterostructure is found to be modulated by the twist angle between the atomically thin layers, elucidating the effect of moire potentials on exciton motion and providing guidelines to design quantum photonics devices based on 2D materials. The nanoscale periodic potentials introduced by moire 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 moire potentials is still limited. Here we investigated interlayer exciton dynamics and transport in WS2-WSe2 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 moire potentials around 100 meV and deviates from normal diffusion due to the interplay between the moire 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.