This question is a little old but I want to complement the multiple great answers given above with some more similarities between the ``classical'' Fourier transform and the Fourier-Mukai transform.

**Compositions**

Let us denote by $\phi$ the ``classical'' Fourier transform. It has already been mentioned above that $\phi^4$ is the identity, but it has not been mentioned yet that $\phi^2(f) = f(-x)$. 

Let us denote by $\Phi_{\mathcal P}$ the Fourier-Mukai transform on the derived category of coherent sheaves on an abelian variety $X$ and its dual $\hat X$. So $\Phi_{\mathcal P}$ sends a complex of coherent sheaves on $X$ to a complex of coherent sheaves on $\hat X$. It can be proved that $\Phi_{\mathcal P}^4$ is the identity and moreover $\Phi_{\mathcal P}^2 = -1_X^*[-g]$. Here $-1_X : X \to X $ denotes the inverse morphism of $X$ and $[-g]$ denotes a shift by $-g = -\dim X$.

So the above shows that compositions of $\Phi_{\mathcal P}$ behave roughly ``the same'' as compositions of $\phi$.

**Product and Convolution**

Another striking similarity is how $\phi$ and $\Phi_{\mathcal P}$ handle products and convolutions. Recall that for functions $f$ and $g$ satisfying certain requirements we can define their convolution 
$$ f * g (x) := \int_{-\infty}^{\infty} f(t)g(t - x) \text{d}t. $$

A well known theorem then states that $\phi(f * g) = \phi(f) \cdot \phi(g)$ and $\phi(f \cdot g) = \phi(f) * \phi(g)$.

We can also define a sort of convolution for coherent sheaves on abelian varieties. Suppose $\mathcal F, \mathcal G$ are coherent sheaves on $X$ and $\hat X$ respectively. We define the convolution of these sheaves as
$$ \mathcal F * \mathcal G := \mu^*(p_X^*\mathcal F \otimes p^*_{\hat X}\mathcal G ). $$
Here $\mu : X \times_k X \to X$ denotes the group law of the abelian variety $X$. 

Now it can be shown that $\Phi_{\mathcal P}(\mathcal F * \mathcal G) = \Phi_{\mathcal P}(\mathcal F) \otimes \Phi_{\mathcal P}(\mathcal G)$ and $\Phi_{\mathcal P}(\mathcal F \otimes \mathcal G) = \Phi_{\mathcal P}(\mathcal F) * \Phi_{\mathcal P}(\mathcal G)[g]$