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In the spirit of the classical theory for K3 surfaces, we establish a modulitheory for families of marked K3 surfaces over smooth rational curves in the K3period domain. In particular, we study the geometry of the space of suchrational curves, which assumes the role of a period space for this moduliproblem. Over an open subset containing all twistor cycles, we construct afamily of such families that is a universal small deformation for every twistorfamily. Finally, small deformations of K3 twistor families are shown to inducecomplex-hyperk\"ahler metrics on members of the families via the originalcomplex version of Penrose's Non-linear Graviton construction. This answers aquestion posed independently by Fels--Huckleberry--Wolf and Looijenga.Generalizations to higher-dimensional irreducible holomorphic-symplecticmanifolds are studied throughout.

Moduli of K3 families over $\mathbb{P}^1$, cycle spaces of IHS period domains, and deformations of complex-hyperk\"ahler metrics