Optical quasiparticles called magnetic excitons recently emerged in magnetic van der Waals materials. Akin to the highly effective strategies developed for electrons, the strong interactions of these excitons with the spin degree of freedom may provide innovative solutions for long-standing challenges in optics, such as steering the flow of energy and information. Here we demonstrate the transport of excitons by spin excitations in the van der Waals antiferromagnetic semiconductor CrSBr. Our observations reveal ultrafast, nearly isotropic exciton propagation, substantially enhanced at the Néel temperature, transient contraction and expansion of exciton clouds at low temperatures and superdiffusive behaviour in bilayer samples. These signatures largely defy description by commonly known exciton transport mechanisms. Instead, we attribute them to magnon currents induced by laser excitation. We propose that the drag forces exerted by these currents can effectively imprint characteristic properties of spin excitations onto the motion of excitons. The universal nature of the underlying magnon–exciton scattering promises the driving of excitons by magnons in other magnetic semiconductors and even in non-magnetic materials by proximity in heterostructures, merging the rich physics of magnetotransport with optics and photonics.