Inequality for Lévy SDE

Question

Let $X_{s}^{t,x}$ denote the solution at time $s$ of an Ito SDE whose coefficients are Lipschitz continuous with initial condition $X_t=x$. Let $t\leq s\leq T<\infty$. The inequality $$ \mathbb{E}\left[\left|X_{s}^{t,x}-X_{s}^{t,\hat{x}}\right|^{2}\right]\leq C\left|x-\hat{x}\right|^2\text{ for }s\in[t,T] $$ is well-known and established by Grönwall's lemma. Are there are any nontrivial cases where such an inequality, or anything implying some kind of uniform continuity (e.g. Hölder, Lipschitz, etc.) holds for a Lévy SDE?