  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 \'etale
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-hyperk\"ahler
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.
