Let $Q$ be a Markov kernel on $([0,1),\mathcal B([0,1))$ nice enough to ensure that $$\kappa(x,\;\cdot\;):=\bigotimes_{n\in\mathbb N_0}Q(x_n,\;\cdot\;)\;\;\;\text{for }x\in[0,1)^{\mathbb N_0}$$ is a well-defined Markov kernel on $\left([0,1)^{\mathbb N-0},{\mathcal B([0,1))}^{\otimes\mathbb N_0}\right)$. Assume $U,V$ are $[0,1)^{\mathbb N_0}$-valued random variables such that $(U,V)\sim\mathcal L(U)\otimes\kappa$, i.e. $$\operatorname P\left[V\in B\mid U\right]=\kappa(U,B)\;\;\;\text{almost surely for all }B\in{\mathcal B([0,1))}^{\otimes\mathbb N_0}\tag1.$$
Now let $k\in\mathbb N$, $S:=\lfloor kU_0\rfloor$ and $T:=\lfloor kV_0\rfloor$. How can we find the joint distribution of $(S,(U_n)_{n\in\mathbb N})$ and $(T,(V_n)_{n\in\mathbb N})$?
Let $$\pi_I:[0,1)^{\mathbb N}\to[0,1)^I\;,\;\;\;x\mapsto(x_i)_{i\in I}$$ for $I\subseteq\mathbb N$. It should be sufficient to consider the distributions of sets of the form $\{i\}\times\pi_I^{-1}\left(\times_{i\in I}A_i\right)\times\{j\}\times\pi_J^{-1}\left(\times_{j\in J}B_j\right)$ for $i,j\in\{0,\ldots,k-1\}$ and $(A_i)_{i\in I},(B_j)_{j\in J}\subseteq\mathcal B([0,1))$ for some finite nonempty $I,J\subseteq\mathbb N$. Given such a set, let $I_0:=\{0\}\cup I$, $J_0:=\{0\}\cup J$, $$A_0:=\left[\frac ik,\frac{i+1}k\right)$$ and $$B_0:=\left[\frac jk,\frac{j+1}k\right).$$ We should have \begin{equation}\begin{split}&\operatorname P\left[((S,(U_n)_{n\in\mathbb N}),(T,(V_n)_{n\in\mathbb N}))\in\{i\}\times\pi_I^{-1}\left(\times_{i\in I}A_i\right)\times\{j\}\times\pi_J^{-1}\left(\times_{j\in J}B_j\right)\right]\\&=\operatorname P\left[(U,V)\in\pi_{I_0}^{-1}\left(\times_{i\in I_0}A_i\right)\times\pi_{I_0}^{-1}\left(\times_{j\in J_0}B_j\right)\right]\\&=\prod_{i\in I_0\setminus J_0}\operatorname P\left[U_i\in A_i\right]\cdot\prod_{j\in I_0\cap J_0}\int_{A_j}\operatorname P\left[U_j\in{\rm d}u_j\right]Q(u_j,B_j)\cdot\prod_{j\in J_0\setminus I_0}\int\operatorname P\left[U_j\in{\rm d}u_j\right]Q(u_j,B_j)\end{split}\end{equation}