A correct statement would be : suppose Y is unramified/étale/smooth over X, then $Y\times_X Z\to X$ is unramfied/étale/smooth iff $Z\to X$ is.

A correct statement would be : suppose Y is unramified/étale/smooth over X, then $Y\times_X Z\to X$ is unramfied/étale/smooth iff $Z\to X$ is.

[EDITED] According to Matt's remark below, we must suppose $Y\to X$ surjective.