Why is a random walk on $\mathbb{R}^d$ (see below) a time-homogeneous Markov process? Specifically, why does it satisfy requirement #2 of definition 17.3 that the map $\left(x,B\right)\mapsto\mathrm{P}_x\left[X\in B\right]$ be a stochastic kernel?
Relevant Definitions
Let $Y_1,Y_2,\dots$ be i.i.d.$\mathbb{R}^d$-valued random variables and let $S_n^x=x+\sum_{i=1}^n Y_i\space\space \mathrm{for}\space x\in\mathbb{R}^d\space\mathrm{and}\space n\in\mathbb{N}_0$
Define probability measures $\mathrm{P}_x$ on $\left(\left(\mathbb{R}^d\right)^{\mathbb{N}_0},\left(\mathcal{B}\left(\mathbb{R}^d\right)^{\otimes\mathbb{N}_0}\right)\right)$ by $\mathrm{P}_x=\mathrm{P}\circ\left(S^x\right)^{-1}$. Then the canonical process $X_n:\left(\mathbb{R}^d\right)^{\mathbb{N}_0}\rightarrow\mathbb{R}^d$ is a Markov chain with distributions $\left(\mathrm{P}_x\right)_{x\in\mathbb{R}^d}$. The process $X$ is called a random walk on $\mathbb{R}^d$ with initial value $x$. [Klenke, Example 17.5, p. 347]