We consider two properties of the Kontsevich moduli spaces of genus-0 stable maps to a variety X. The first, irreducibility, implies that certain genus-0 Gromov-Witten invariants are enumerative. The second, existence of very twisting families, implies the existence of sections for a two parameter family with vanishing elementary obstruction. Both of these properties are known to hold for homogeneous varieties, as well as low degree hypersurfaces in projective space.Motivated by these results for projective space, we prove that the Kontsevich moduli spaces are irreducible when X is a low degree hypersurface in a Grassmannian variety. We conjecture a sharp inequality kd^2 < n for when a two parameter family of degree $d$ hypersurfaces in the Grassmannian G(k,n) with vanishing elementary obstruction admits a rational section, and prove that a slightly weaker result holds.