 Solution

Failure in general. Let $f:{\mathbb{A}}_k^1=\mathrm{Spec}k[t]\to \mathrm{Spec}k$ be the structure morphism and take the trivial line bundle $\mathcal{L}={\mathcal{O}}_{{\mathbb{A}}_k^1}$. The polynomial ring $k[t]$ is a Noetherian ring, so this is a coherent sheaf. Its direct image on the one-point target has sectionsA coherent sheaf on $\mathrm{Spec}k$ corresponds to a finite-dimensional vector space over $k$, whereas $1,t,t^2,\dots$ are linearly independent. Thus

Quasi-coherence for the stated morphism. Now let $f$ satisfy (a), and let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Over any affine open subscheme $V=\mathrm{Spec}A$ of $Y$, write $f^{-1}V=\mathrm{Spec}B$ and $\mathcal{F}{|}_{f^{-1}V}=\tilde{M}$ for a $B$-module $M$. Regard $M$ as an $A$-module through the ring homomorphism $A\to B$. For $a\in A$,The last two localizations are identical: both invert the action of the same element on $M$. These identifications commute with restrictions, so on the basis of principal open subsets they identify $(f_{\ast }\mathcal{F}){|}_V$ with the affine module sheaf $\widetilde{{}_{A}M}$. Hence this direct image is a quasi-coherent sheaf on each $V$, and therefore on $Y$. This part uses only that $f$ is an affine morphism.

Coherence under the Noetherian hypotheses. If $X,Y$ are Noetherian schemes and $\mathcal{F}$ is a coherent sheaf, then $M$ is a finitely generated module over $B$. Choose generators $m_1,\dots ,m_s$ of $M$ over $B$, and generators $b_1,\dots ,b_r$ of $B$ over $A$, supplied by (a). Then the finite familygenerates $M$ as an $A$-module: expand each coefficient of an expression $m={\sum }_{\ell }{c}_{\ell }{m}_{\ell }$ in the $b_j$. Since $A$ is a Noetherian ring, the affine module sheaf $\widetilde{{}_{A}M}$ is a coherent sheaf. This holds on an affine open cover of $Y$, proving