ag.algebraic geometry - Fourier-Mukai transform - a first example

Indeed, as follows from the comments below, maps between schemes provide examples of Fourier-Mukai transform, most famous example being a similar map with additional twisting by a bundle in $Atimes hat A$ for an Abelian variety $A$.

Anyway, since the restriction $p'_X:Gamma_fto X$ is actually an isomorphism (the inverse is $xmapsto (x, f(x))$) and the composition $p_Ycirc {p'_X}^{-1}: X to Gamma_f to Y$ is exactly $f$, the statement you have written is actually equivalent to $f_* () = f_*()$. Thus there is no hard commutative algebra stuff.