Let $E$ be a separable $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous, $$\rho(x,y):=\inf_{\substack{\gamma\:\in\:C^1([0,\:1],\:E)\\ \gamma(0)\:=\:x\\ \gamma(1)\:=\:y}}\int_0^1v\left(\gamma(t)\right)\left\|\gamma'(t)\right\|_E\:{\rm d}t\;\;\;\text{for }x,y\in E,$$ $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space, $X:\Omega\times[0,\infty)\times E\to E$ be a stochastic flow, $$X^x_t:=X(\;\cdot\;,t,x)\;\;\;\text{for }(t,x)\in[0,\infty)\times E$$ and $$\kappa_t(x,B):=\operatorname P\left[X^x_t\in B\right]\;\;\;\text{for }(x,B)\in E\times\mathcal B(E)\text{ and }t\ge0.$$

Assume $$\operatorname E[v(X^x_t)]\le cv^{\lambda(t)}(x)\;\;\;\text{for all }(t,x)\in[0,\infty)\times E\tag1$$ for some $c>0$ and decreasing $\lambda:[0,\infty)\to[0,1]$. By $(1)$, $$\operatorname E[\rho(X^x_t,X^y_t)]\le c\rho(x,y)\tag2$$ for all $x,y\in E$ and $t\in[0,1]$.

Let, $\mathcal M_1$ denote the set of probability measures on $(E,\mathcal B(E))$, $\operatorname W_\rho$ denote the Wasserstein metric associated with $\rho$ and $$\mathcal S^1:=\{\mu\in\mathcal M_1\mid\exists y\in E:(\mu\otimes\delta_y)\rho<\infty\}.$$ By $(2)$, $$\operatorname W_\rho(\delta_x\kappa_t,\delta_y\kappa_t)\le c\operatorname W_\rho(\delta_x,\delta_y)\tag3$$ for all $x,y\in E$ and $t\in[0,1]$.

Let $t\ge0$. Can we show that $\kappa_t^\ast$ is $\mathcal S^1$-preserving? Or even that $\kappa_t^\ast\mathcal M_1\subseteq\mathcal S^1$?

I'm quite sure that at least the $\mathcal S^1$-preserving claim is true. If $\mu\in\mathcal M_1$, then we need to show that there is a $y\in E$ with $(\mu\kappa_t\otimes\delta_y)\rho_r<\infty$. Maybe we can pick $y=0$.

**EDIT 1**: Assume $\delta_x\kappa_t\in S^1$ for all $x\in E$ and $t\ge0$.

**EDIT 2**: Assume there are nondecreasing $v_i:[0,\infty)\to(1,\infty)$ with $v_1(\left\|x\right\|_E)\le v(x)\le v_2(\left\|x\right\|_E)$ for all $x\in E$ and $rv_2(r)\le \alpha v_1^\beta(r)$ for all $r>0$ for some $\alpha\ge0$ and $\beta\ge1$. Assume further that $\operatorname E[V^\theta(X^x_t)]\le\eta v^{\beta\lambda(t)}(x)$ for all $x\in E$ and $t\ge0$.

Then we easily see $\rho(0,x)\le\alpha v^\beta(x)$ for all $x\in E$. Now, since $\lambda$ is decreasing, it must hold $\lambda(t)\to0$ as $t\to\infty$ and hence $$\operatorname W_\rho(\mu\kappa_t,\delta_0)=\int\mu({\rm d}x)\operatorname E[\rho(0,X^x_t)]\le\alpha\eta\int\mu({\rm d}x)v^{\beta\lambda(t)}(x)\xrightarrow{t\to\infty}1\tag4$$ by monotone convergence for all $\mu\in\mathcal M_1$ and $t\ge0$. This should yield that $\kappa_t^\ast$ maps $\mathcal M_1$ to $\mathcal S^1$ for all $t\ge0$.