Transition kernels of diffusions: kernel properties

Question

Question: Suppose $Y_{t,x}$ is an Ito diffusion, i.e. the a.s. unique solution of the SDE $$ dY_{t, x} = b(Y_{t, x})dt + \sigma(Y_{t, x})dW_t, \qquad Y_{0, x}=x, $$ with $\sigma, b$ Lipschitz continuous. We can define the family of "transition functions" $(P_t)_{t\geq0}$ of the process via $$ P_t: \mathbb{R}^d \times \mathbb{B}(\mathbb{R}^d) \to [0, 1], (x, B) \mapsto \mathbb{P}(Y_t\in B | Y_0=x). $$ How do we know, that $P_t$ is a transition kernel for each $t$, i.e. that $$ P_t(\cdot, B): x \mapsto P_t(x, B) $$ is measurable for all $B\in\mathbb{B}(\mathbb{R}^d)$ and $$ P_t(x, \cdot): B \mapsto P_t(x, B) $$ is a probability measure for all $x\in\mathbb{R}^d$? What assumptions do we need for that?
Own effort: I was trying to show the semigroup property of the transition semigroup $(P_tf)(x):=\mathbb{E}[f(Y_{t,x})]$, which can be traced back to the Chapman-Kolmogorov Equation, which in turn only makes sense for transition kernels, i.e. if the defined transition functions have the kernel properties. However, this is often just assumed/stated as for example in [Pavliotis, Stochastic Processes and Applications, p.35]. Is this obvious? I could understand the second part, i.e. that $P_t(x,\cdot)$ is a probability measure, since that might just be a consequence of there being a solution to the SDE for every x?