  We describe the induced geometry on several classes of Kodaira moduli spaces
of rational curves in twistor spaces. By constructing connections and frames on
the moduli spaces we build and review twistor theories pertaining to
relativistic and non-relativistic geometries. Focussing on the cases of three-
and five-dimensional moduli spaces we establish novel twistor theories of
Newton-Cartan spacetimes. We generalise a canonical class of connections on
Kodaira moduli spaces to encompass torsion and prove that in three dimensions
deformations of the twistor space's holomorphic structure induce Newton-Cartan
structures with torsion. In five dimensions the canonical connections contain
generalised Coriolis forces, and here also we consider the introduction of
torsion by deformation. Non-relativistic limits are exhibited as jumping
phenomena of normal bundles. We consider jumping phenomena in Newtonian twistor
theory, as well as exhibiting via twistor theory that three-dimensional
torsional Newton-Cartan geometry generically exists on the jumping
hypersurfaces of Gibbons-Hawking manifolds.
