Let $(X_t)_{t \in \mathbb N}$ be a stochastic process on a probability space $(\Omega, \mathcal F, \mathbb P)$ with countable state space $S$.
Let $n \ge 1, t_1<t_2<\ldots <t_{n+1} \in \mathbb N$ and $A_1,\ldots,A_{n+1}\subseteq S$ such that $P[X_{t_n} \in A_n, \ldots X_{t_1} \in A_1]>0$.
Consider
$$\mathbb P[X_{t_{n+1}} \in A_{n+1}|X_{t_n} \in A_n, \ldots , X_{t_1} \in A_1]=\mathbb P[X_{t_{n+1}} \in A_{n+1}|X_{t_n} \in A_n]$$
If $X_1,X_2,X_3,\ldots$ are independent, then this is true.
I'm looking for a counter-example to disprove the other direction, i.e.
$$\mathbb P[X_{t_{n+1}} \in A_{n+1}|X_{t_n} \in A_n, \ldots , X_{t_1} \in A_1]=\mathbb P[X_{t_{n+1}} \in A_{n+1}|X_{t_n} \in A_n]$$ but $X_1,X_2,X_3,\ldots$ are not independent