  We show that for many moduli spaces M of torsion sheaves on K3 surfaces S,
the functor D(S) -> D(M) induced by the universal sheaf is a P-functor, hence
can be used to construct an autoequivalence of D(M), and that this
autoequivalence can be factored into geometrically meaningful equivalences
associated to abelian fibrations and Mukai flops. Along the way we produce a
derived equivalence between two compact hyperkaehler 2g-folds that are not
birational, for every g >= 2. We also speculate about an approach to showing
that birational moduli spaces of sheaves on K3 surfaces are derived-equivalent.
