  We study certain sequences of moduli spaces of sheaves on K3 surfaces,
building on work of Markman, Yoshioka, and Nakajima. We show that these
sequences can be given the structure of a geometric categorical sl_2 action in
the sense of Cautis, Kamnitzer, and Licata. As a corollary, we get an
equivalence between derived categories of some moduli spaces that are
birational via stratified Mukai flops.
