Eosinophils and basophils are broadly referred to as non-professional phagocytes, in that they _can_ use phagocytosis, but don't primarily function like this. Rather, they tend to de-granulate to kill their targets. They lack the abundance of receptor complexes for efficient phagocytosis that professional phagocytes like monocytes and neutrophils express, like opsonin receptors.
There's some data to suggest that macrophages control non-professional phagocytes like epithelial and endothelial cells (ref), but it's a bit of a jump to generalize that to eosinophils and basophils.
My intuition is that these specialized types of cells uptake debris and extracellular antigen, since they do partake in non-professional class-II antigen presentation, rather than larger bodies or microorganisms like macrophages uptake.