I am a little confused about what I think must be either a standard theorem or a standard counterexample in model theory, and I am hoping that the MathOverflow model theorists can help sort me out about which way it goes.

My situation is that I have a chain of submodels, which is not necessarily an elementary chain,

$$M_0\subseteq M_1\subseteq M_2\subseteq\cdots$$

and I have elementary embeddings $j_n:M_n\to M_n$, which cohere in the sense that $j_n=j_{n+1}\upharpoonright M_n$. So there is a natural limit model $M=\bigcup_n M_n$ and limit embedding $j:M\to M$, where $j(x)$ is the eventual common value of $j_n(x)$.

Question. Is the limit map $j:M\to M$ necessarily elementary?

A natural generalization would be a coherent system of elementary embeddings $j_n:M_n\to N_n$, with possibly different models on each side. The question is whether the limit embedding $j:M\to N$ is elementary, where $M=\bigcup_n M_n$, $N=\bigcup_n N_n$ and $j=\bigcup_n j_n$.

I thought either there should be an easy counterexample or an easy proof, perhaps via Ehrenfeucht-Fraïssé games?