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 $P[X_{t_n} \in A_n, \ldots X_{t_1} \in A_1]>0$.
If $X_1,X_2,X_3,\ldots$ are independent, then we have:
$$\mathbb P[X_{t_{n+1}} \in A_{n+1}|X_{t_n} \in A_n, \ldots , X_{t_1} \in A_1]=\mathbb P[X_{t_{n+1}} \in A_{n+1}|X_{t_n} \in A_n]$$
I'm looking for a counterexample to disprove the other direction.
Actually, what you have written is a Markov property of stochastic process, its not independence. If the random variables $X$ and $Y$ are independent then you have $$ \mathbb{P}(X \in A ~|~ Y \in B) = \mathbb{P}(X \in A). $$
Often people assume Markov property, since they know that the process $(X_t)_{t\in\mathbb{N}}$ is not independent and Markov property makes life a little bit easier.