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

In the spirit of the classical theory for K3 surfaces, we construct a Hausdorff fine moduli space for families of marked K3 surfaces over smooth rational curves in the K3 period domain. This moduli space admits an étale holomorphic map to the parameter space for such rational curves, which assumes the role of a period space for this moduli problem. Over an open subset containing all twistor cycles, we construct a family of such families that is a universal small deformation for every twistor family. Finally, small deformations of K3 twistor families are shown to induce complex-hyperkähler metrics on members of the families via the original complex version of Penrose's Non-linear Graviton construction. This answers a question posed independently by Fels--Huckleberry--Wolf and Looijenga. Generalizations to higher-dimensional irreducible holomorphic-symplectic manifolds are studied throughout.