Here is a result whose proof uses Fourier-Mukai duality:
Consider a family of abelian varieties $A \rightarrow X$, its dual $\check{A} \rightarrow X$, and a torsor $\mathcal{T}$ (for $A \rightarrow X$) and a gerbe $\mathcal{G}$ on $\mathcal{T}$.
Then there exists a "dual torsor" $\check{\mathcal{T}} \rightarrow X$ (for $\check{A} \rightarrow X$), and a "dual" gerbe $\check{\mathcal{G}}$ on $\check{\mathcal{T}}$, such that the derived categories of coherent sheaves on $\mathcal{G}$ and $\check{\mathcal{G}}$ are equivalent.
My question is how to define the dual "gerbe" and its dual "torsor". My rough understanding is that the dual "torsor" $\check{\mathcal{T}}$ is defined as follows: a point in $\check{\mathcal{T}}$ corresponds to a splitting of the restriction of the gerbe $\mathcal{G}$ to a fiber of $\mathcal{T} \rightarrow X$. I'm not sure about how to construct the dual "gerbe".
