Many mathematicians have worked on the problem of determining which homogeneous spaces $G/H$ admit proper and cocompact actions by discrete groups $\Gamma$. This question is highly nontrivial when $H$ is noncompact, and still far from being solved. I will consider homogeneous spaces $G/H$ that do admit such actions and examine the deformation of the compact quotients $\Gamma/G/H$. I will prove that for most known examples with $G$ and $H$ reductive, the proper discontinuity of the action is preserved under any small deformation of $\Gamma$ in $G$. For $G/H=SO(2,2)/SO(1,2)$, this is related to the existence of Thurston's asymmetric distance on Teichmuller space. I will also address similar questions in the setting of $p$-adic homogeneous spaces.