Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE \begin{equation} dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,. \end{equation}
Let $f(x,t|x_0,0)$ denote its transition density function. Under suitable conditions, as is my case, we know that $f$ is Lipschitz on every relatively compact set $S$ with associated constant $C_S$.
Due to the Markovian property, we have that \begin{equation} f(x_1,t_1;x_2,t_2;\ldots;x_n,t_n|x_0,t_0) = f(x_n,t_n|x_{n-1},t_{n-1}) \ldots f(x_2,t_2|x_{1},t_{1}) f(x_1,t_1|x_0,t_0)\,. \end{equation}
Does this imply that the finite dimensional distribution is Lipschitz on $S^n$ as well (in terms of $x=(x_1,\ldots,x_n)$)? And, if so, can such a constant be explicitly determined as well in terms of some $C_S$'s?
Edit 1:
By the finite-dimensional density being Lipschitz, I meant that for every relatively compact set $S$ there is a constant $C_S$ such that
\begin{equation} |f(x_n,\ldots,x_1|x_0) - f(y_n,\ldots,y_1|x_0)| \leq C_S |x-y| \end{equation} with $x=(x_1,\ldots,x_n)\,, y=(y_1,\ldots,y_n)$. Note I omitted the times for simplicity.
