Dark matter substructures emerge naturally in axion cosmology. The DALI experiment is a next-generation interferometer for wavelike dark matter in the 25–250 μeV range, currently in its prototyping phase. Although DALI's main objective is to explore the dark matter in our galaxy as a haloscope, to a large degree in this article we explore the prospect for detection of fine grained streams made of axions that would enter the Solar System, as suggested by previous work, which would be performed in parallel without interfering with the search for virialized axions. We find that DALI will have a sensitivity to encounters with these coherent streams that fully spans the window defined by the coupling strength to photons of representative axion models over a stacked bandwidth of two decades in mass.