To answer Martin's questions in his comments to Rayleigh's answer:
Fix a scheme $S$ and a proper morphism $f:X\to Y$ of $S$-schemes. Suppose that $Y$ is proper over $S$. If $f$ is proper, then $X$ is proper over $S$. This is simply because proper morphisms are stable under composition.
How does one prove that proper morphisms are stable under composition? One simply proves this property for finite type morphisms of schemes and separated morphisms. Then you're done.
If you stick to fields, all morphisms are separated so to prove that proper morphisms are stable under composition in this case, you simply have to prove that if you have a tower $K\subset L\subset M$ of finite degree field extensions, then $K\subset M$ is of finite degree. This is an easy fact. In conclusion, the proof of the statement for fields is easy and the statement itself doesn't give any nontrivial information in the case of fields.
I think Rayleigh added the additional hypotheses of "finite etale" to his statement, because he wanted to mimic the set-up of a "covering".