There is a notion of "deformation in the large", at least for smooth projective varieties. A deformation in the large is any projective, $\textit{smooth}$ morphism over a connected base. Your question does make sense for deformations in the large. There are a number of varieties which are known to be rigid for deformations in the large, i.e., the morphism is a product étale locally over the base. The first example is projective space (which follows, e.g., from Mori's solution of the Hartshorne conjecture). I believe Siu first proved that hyperquadrics are rigid for deformations in the large. And there has recently been phenomenal work of Jun-Muk Hwang and Ngaiming Mok establishing the same result for any irreducible Hermitian symmetric space, e.g., Grassmannians.