Let $f:X \to Y$ be a finite, faithfully flat morphism of noetherian, affine $\mathbb{C}$-schemes. One can assume $Y$ is non-singular. Let $A$ be a local artinian $\mathbb{C}$-algebra and $f_A:X_A \to Y_A$ be the trivial deformation of $f$, where $X_A:=X \times \mbox{Spec}(A)$, $Y_A:=Y \times \mbox{Spec}(A)$ and $f_A$ is simply $f \times \mbox{id}$. Let $\mathcal{F}_A$ be a coherent sheaf on $X_A$, which is $A$-flat. Is $(f_A)_*\mathcal{F}_A$ going to be $A$-flat?

Yes, it follows from the projection formula https://stacks.math.columbia.edu/tag/08EU For any $A$-module $M$ we have $$Rf_{A*}\mathcal{F}_A\otimes^{L}_{\mathcal{O}_Y}p^*M\simeq Rf_{A*}(\mathcal{F}_A\otimes^L_{\mathcal{O}_X}Lf_A^*(p^*M))$$ where $p:Y_A\to Spec\, A$ is the structure morphism. Since $f_A$ is flat, $Lf_A^*(p^*M)=q^*M$ where $q:X_A\to Spec\, A$ is the structure morphism of $X_A$ and the RHS of the projection formula is the derived pushforward of a quasi-coherent sheaf $\mathcal{F}_A\otimes_{\mathcal{O}_X}f_A^*p^*M$(we can replace the derived tensor product by the usual tensor product since $\mathcal{F}_A$ is $A$-flat) along the affine morphism $f_A$, so it is concentrated in degree 0.

In other words, this shows that the derived tensor product of $f_{A*}\mathcal{F}_A=Rf_{A*}\mathcal{F}_A$ with any $A$-module is concentrated in degree $0$. It means that $f_{A*}\mathcal{F}_A$ is $A$-flat.