Let $\iota_\ast : \mathcal A \to \mathcal B$ be a geometric morphism. I'm looking for some functor
$$F_{\mathcal A \to \mathcal B} : \mathrm{Topos}_{//\mathcal B} \to \mathrm{Topos}_{//\mathcal A}$$
which can be defined from the data of $\iota$. When $\mathcal B = \mathrm{BRing} = \operatorname{Fun}(\mathrm{Ring}^\text{fp}, \mathrm{Set})$ is the classifying topos for (commutative) rings, so that $\mathrm{Topos}_{//\mathcal B}$ is the category of ringed topoi, I'd like the following to be true when the input $\langle R \rangle_\ast : \mathrm{Set} \to \mathrm{BRing}$ classifies an ordinary ring $R$:
$F_{\mathrm{Zar} \to \mathrm{BRing}}(R) = \operatorname{Spec}^{\mathrm{Zar}}(R) \xrightarrow{\langle \mathcal O_R\rangle_\ast} \mathrm{Zar}$.
$F_{\mathrm{Et} \to \mathrm{BRing}}(R) = \operatorname{Spec}^\text{ét}(R) \xrightarrow{\langle \mathcal O_R\rangle_\ast} \mathrm{Ét}$.
That is, when $\mathcal A$ is the classifying topos for local rings, I want to carry a ring to its Zariski spectrum (with the usual structure sheaf of local rings), and when $\mathcal A$ is the classifying topos for strictly henselian local rings, I want to carry a ring to its étale spectrum.
Question: Does such a general construction $\iota \mapsto F_\iota$ exist?
Notes:
This should be related to Cole's theory of spectrum, or maybe to Lurie's theory of spectrum, but in these cases it seems the input data is a bit different?
Of course, I'd expect this functor to carry morphisms to morphisms of locally ringed topoi in case (1), which are not the same as the morphisms in $\mathrm{Topos}_{/{\mathrm{Zar}}}$, but what I'm asking for should still be perfectly well-defined, even if one can say more. An answer shedding light on what more can be said about the output morphisms would be nice, but not necessary for me.
By $\mathrm{Topos}_{//\mathcal B}$ I mean the lax slice category (which the nlab calls the "slice 2-category") of the 2-category $\mathrm{Topos}$. (Note that if we used $\mathrm{Topos}_{/\mathcal B}$ instead, then we would not have a copy of the category of rings sitting inside when $\mathcal B = \mathrm{BRing}$, just its underlying groupoid.)