Twisting derived equivalences

We introduce a new method for twisting relative equivalences of derived categories of sheaves on two spaces over the same base. The derived categories of sheaves on the spaces are twisted to derived categories of sheaves on gerbes living over spaces that are locally (on the base) isomorphic to the original spaces. This is done in a compatible way so that the equivalence is maintained. We apply this method by proving the conjectures of Donagi and Pantev on dualities between gerbes on genus-one fibrations and comment on other applications