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We study certain sequences of moduli spaces of sheaves on K3 surfaces, building on work of Markman [J. Algebraic Geom. 10 (2001), pp. 623–694], Yoshioka [J. Reine Angew.Math. 515 (1999), pp. 97–123], and Nakajima [ Convolution on homology groups of moduli spaces of sheaves on K3 surfaces , Contemp. Math., vol. 322, Amer. Math. Soc., Providence, RI, 2003, pp. 75–87]. We show that these sequences can be given the structure of a geometric categorical \mathfrak {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. [ABSTRACT FROM AUTHOR]