Derived equivalences of gerbey curves

We study derived equivalences of certain stacks over genus $1$ curves, which arise as connected components of the Picard stack of a genus $1$ curve. To this end, we develop a theory of integral transforms for these algebraic stacks. We use this theory to answer the question of when two stacky genus $1$ curves are derived equivalent. We use integral transforms and intersection theory on stacks to answer the following questions: if $C^{\mathrm{\prime }}=Pi{c}^d(C)$, is $C=Pi{c}^f(C^{\mathrm{\prime }})$ for some integer $f$? If $C^{\mathrm{\prime }}=Pi{c}^d(C)$ and $C^{\mathrm{\prime }\mathrm{\prime }}=Pi{c}^f(C^{\mathrm{\prime }})$, then is $C^{\mathrm{\prime }\mathrm{\prime }}=Pi{c}^g(C)$ for some integer $g$?