Unramification stable under change base

It's easier to think about unramified as finitely presented and $\Omega_{X/Y}^1$ vanishing. It encodes what you're trying to do into one neat package. So, now suppose that $f:X\to Y$ is unramified, and $g:Z\to Y$ is some morphism. Consider the diagram

$$\begin{matrix}X\times_Y Z & \xrightarrow{p} & X\\ \downarrow & & \downarrow^f \\ Z & \xrightarrow{g} & Y\end{matrix}$$

Then, it's easy to show that the pullback is finitely presented, and that $\Omega^1_{X\times_Y Z/Z}=p^\ast(\Omega_{X/Y}^1)$. But, it clearly follows then that if $\Omega^1_{X/Y}=0$ then so does $\Omega^1_{X\times_Y Z/Z}$.

Or, it's also easier in this context to think about unramified as having your diagional map be an open embedding. Then the claim is clear from the fact that open embeddings are invariant under base change and setting up the right fibered diagram to relate the diagionals of the original map and the base change's diagional.