A linear operator on Wasserstein spaces

Question

Let $(S,d)$ be a Polish space, and let $p\in[1,\infty)$. We define $$P=\left\{\mu:\mu \text{ is a probability measure on } S \text{ s.t. } \int_Sd(x,y)^pdy<\infty \text{ for every } x\in S\right\}.$$ Let $\pi$ be a transition kernel on $S$, and let $\mu\in P$. How can we show $\pi\mu\in P$, where $$\pi\mu(B)=\int_S\pi(x,B)\mu(dx)?$$