Cross stratification occurs due to a depositional geological process. In this case the process would be a paleocurrent moving ripples or dunes downstream. Deltas typically have one or more flow channels and will thus deposit geological flow features such as cross bedding. Given the close proximity of the sediment source, the grain size is expected to be large, as you said. The smaller grains were likely entrained within the high energy currents that was causing the ripple and dune migration and deposited towards the distal portions of the delta as current energy waned. Deltas are very complex depositional environments and can have a wide variety of sorting, grain size, and depositional features within a single delta. These depend on the samples proximity to a deltaic channel, the proximity to the sediment source, and the degree of sediment reworking by waves, tides, and alongshore currents.