Deformations of standard locally homogeneous spaces

Let $X=G/H$ be a homogeneous space, where $G \supset H$ are reductive Lie groups. We ask: in the setting where $\Gamma \backslash G/H$ is a standard quotient in the sense of Kassel–Kobayashi [9], to what extent can the discrete subgroup $\Gamma$ be deformed while preserving the proper discontinuity of the $\Gamma$-action on $X$? We provide several classification results, including: conditions under which local rigidity holds for compact standard quotients $\Gamma\backslash X$; criteria for when a standard quotient can be deformed into a nonstandard one; a characterization of the maximal Zariski-closure of discontinuous groups under small deformations; and conditions under which Zariski-dense deformations occur.