To add a representation theory perspective: if $G$ is a Lie group, and $f$ is a function (or more precisely a distribution) on $G$ then (under certain mild conditions on $f$ and $G$), the function $f$ is uniquely determined by its unitary matrix coefficients, i.e. the coefficients of the matrix $\rho(f)$ where $\rho:G\to GL_n$ goes over all isomorphism classes of unitary and irreducible representations. This perspective should be understood as a change of basis that reveals the "underlying equivariant properties" of a function $f$, i.e. the properties important from the point of view of representation theory.
Now the unitary irreducible representations of the additive group $\mathbb{R}$ are one-dimensional representations $\rho_\alpha: t\mapsto e^{i\alpha t}$ indexed by $\alpha\in \mathbb{R}$, and so the matrix coefficient decomposition of a function is precisely its Fourier transform. This hints that whenever you are interested in problems with additive equivariance (action by $\mathbb{R}$), you should expect to see Fourier transforms.
Your Fourier-Mukai example is an example of the same phenomenon "one category level higher". Namely, coherent sheaves over an algebraic group $G$ form a monoidal category under convolution. A partial analogue of "function $f$ on $G$ acts on line bundles the stack $BG$" (i.e. invertible representations) is "coherent sheaf $F$ on $G$ acts on gerbes on $BG$". In the case of abelian varieties, Gerbes on $BG$ are (more or less) the dual variety and the "matrix coefficients" of this action turn out to precisely be a change of basis (in this case, an equivalence of categories, now given by Fourier-Mukai). For nonabelian groups, the situation is more complicated, since it's not enough to consider gerbes, and it's tricky to say exactly what is an irreducible module category over a monoidal category... but for any reasonable extension of this picture you give there will always be a "matrix coefficient" functor.