I'm having some trouble coming up with a counter-example for this problem:

Give an example of a stochastic process $\{X_n : n \in \mathbb{Z}^+\}$ on $(\Omega, \mathcal{F}, P)$ such that $P_{X_n} = P_{X_0}$ for all $n$ but $\{X_n\}$ is not stationary (I would assume a stationary process?). Here $P_{X_n}$ denotes the pushforward measure of $P$ under $X_n$.

I feel as if I am deeply misunderstanding either the question or the difference of two random variables having the same push-forward measure but bot being iid. Even a hint on the right direction would be much appreciated.

Many thanks in advance!