Let $f:Y\rightarrow X$ be an etale morphism, where $X$ and $Y$ are smooth projective varieties. Let $V$ be a vector bundle over $Y$. Since $f$ is flat, $V$ is flat over $X$. Is it true that $f_*V$ is flat $\mathcal{O}_X$module?
Since $Y$ and $X$ are proper over the base field, the morphism $f$ is finite (by Zariski's main theorem). Hence by the semicontinuity theorem (see Hartshorne p. 288, Cor. 12.9), $f_*V$ is locally free, because $f$ is finite and flat. So the answer is yes. 

