I guess I will just talk about the discrete case and below is my current understanding of things.

Markov chain: A discrete random variable A has several states $A\_{0}, A\_{1}, A\_{2} ... $ over a discrete time t measured in time steps. The probability $T\_{i\rightarrow j}$ of reaching a particular state $A\_{j}$ in next time step depends only on the identity of the current state $A\_{i}$. For example whenever the current state is $A\_{0}$, the probabilities for reaching $A\_{0}, A\_{1}, A\_{2}$ ... are always the same regardless of any past states of A.

Non-stationary process: The probability distribution of states of a discrete random variable A (without knowing any information of current/past states of A) depends on discrete time t. For example, temperature is usually higher in summer than winter. Therefore, the probability distribution of possible temperature over time is a non-stationary random process.

My question is: Can a Markov chain accurately represent a non-stationary process? 

Does a Markov chain always represent a stationary random process?