For any finite group G and any G-C*-algebra A, the collection of all H-equivariant K-theory groups of A assemble to form naturally a Mackey functor k(A). This assignment lifts easily to a stable homological functor k defined on the G-equivariant Kasparov category. We derive from this, and from general homological methods, the existence of previously unknown universal coefficient and Künneth spectral sequences converging to KK^G(A,B) and K^G(A tensor B), respectively. This works well for the G-C*-algebras in a certain equivariant bootstrap category, which is itself of independent interest.