Computing classes of admissible covers

Let Adm(g,h,d) be the space of degree d admissible covers C → D of a genus h curve D by genus g curves C. There is a natural map f : Adm(g,h,d) → Mgnbar into the moduli space of stable curves taking the source curve of an admissible cover and forgetting everything else. When the class [f(Adm(g,h,d))] is tautological we can try to express this class in terms of a known basis for the tautological ring of Mgnbar. We will discuss several strategies for making these computations and give a number of examples.