  We study twisted derived equivalences of hyper-K\"ahler fourfolds. We
describe when two hyper-K\"ahler fourfolds of $K3^{[2]}$-type of Picard rank
$1$ with isomorphic transcendental lattices are derived equivalent. Then we
present new constructions of pairs of twisted derived equivalent hyper-K\"ahler
manifolds of Picard rank $\geq 2$.
