A new paper, written jointly with my Ph.D. student Léo Bartoli, is available from the ArXiv:

The paper carries out a systematic investigation of the functors between sites which induce morphisms between relative toposes.

It culminates in a generalisation of Diaconescu’s theorem for relative toposes, formulated in the language of fibrations and stacks according to the foundations for relative topos theory introduced in the paper Relative topos theory via stacks.