Let me give the relative construction. We'll say our geometric objects are presheaves on some category $Aff$, and we'll denote sheaves of categories by $2QCoh(-)$.

There's a functor

$$P(Aff)_{/B}^{op} \to 2QCoh(B)$$

$$(f: X \to B) \mapsto Perf(X)$$ sending a presheaf over $B$ to a $Perf(X)$ viewed as a sheaf of categories over $B$, via $f^*$. One shows that this preserves limits, and the right adjoint $M_{(-)/B}$ is a relative moduli of objects, as a presheaf over $B$.

As in the absolute case we have an explicit formula

$$M_{C/B}(Spec(R) \to B) \simeq Map_{2QCoh(B)}(Perf(R), C)$$

where in the first argument on the RHS, $Perf(R)$ is endowed with the structure of a sheaf of category over $B$ via the map $Spec(R) \to B$. And of course $B= *$ case recovers the absolute case.

I expect arguments of Antieau-Gepner about representability of moduli generalize to this relative setting, though I do not have a precise statement at the moment.

Footnote: As you do in the question statement I'm sweeping some presentability issues under the rug here to give the idea, $2QCoh(-)$ needs to be constructed carefully, but sounds like you know where to look for these constructions.

Let me give the relative construction. We'll say our geometric objects are presheaves on some category $Aff$, and we'll denote sheaves of categories by $2QCoh(-)$.

There's a functor

$$P(Aff)_{/B}^{op} \to 2QCoh(B)$$

$$(f: X \to B) \mapsto Perf(X)$$ sending a presheaf over $B$ to a $Perf(X)$ viewed as a sheaf of categories over $B$, via $f^*$. One shows that this preserves limits, and the right adjoint $M_{(-)/B}$ is a relative moduli of objects, as a presheaf over $B$.

As in the absolute case we have an explicit formula

$$M_{C/B}(Spec(R) \to B) \simeq Map_{2QCoh(B)}(Perf(R), C)$$

where in the first argument on the RHS, $Perf(R)$ is endowed with the structure of a sheaf of categories over $B$ via the map $Spec(R) \to B$. And of course $B= *$ case recovers the absolute case.

I expect arguments of Antieau-Gepner about representability of moduli generalize to this relative setting, though I do not have a precise statement at the moment.

Footnote: As you do in the question statement I'm sweeping some presentability issues under the rug here to give the idea, $2QCoh(-)$ needs to be constructed carefully, but sounds like you know where to look for these constructions.

Let me give the relative construction. We'll say our geometric objects are presheaves on some category $Aff$, and we'll denote sheaves of categories by $2QCoh(0)$.

There's a functor

$$P(Aff)_{/B}^{op} \to 2QCoh(B)$$

$$(f: X \to B) \mapsto Perf(X)$$ sending a presheaf over $B$ to a $Perf(X)$ viewed as a sheaf of categories over $B$, via $f^*$. One shows that this preserves limits, and the right adjoint $M_{(-)/B}$ is a relative moduli of objects, as a presheaf over $B$.

As in the absolute case we have an explicit formula

$$M_{C/B}(Spec(R) \to B) \simeq Map_{2QCoh(B)}(Perf(R), C)$$

where in the first argument on the RHS, $Perf(R)$ is endowed with the structure of a sheaf of category over $B$ via the map $Spec(R) \to B$. And of course $B= *$ case recovers the absolute case.

I expect arguments of Antieau-Gepner about representability of moduli generalize to this relative setting, though I do not have a precise statement at the moment.

Let me give the relative construction. We'll say our geometric objects are presheaves on some category $Aff$, and we'll denote sheaves of categories by $2QCoh(-)$.

There's a functor

$$P(Aff)_{/B}^{op} \to 2QCoh(B)$$

$$(f: X \to B) \mapsto Perf(X)$$ sending a presheaf over $B$ to a $Perf(X)$ viewed as a sheaf of categories over $B$, via $f^*$. One shows that this preserves limits, and the right adjoint $M_{(-)/B}$ is a relative moduli of objects, as a presheaf over $B$.

As in the absolute case we have an explicit formula

$$M_{C/B}(Spec(R) \to B) \simeq Map_{2QCoh(B)}(Perf(R), C)$$

where in the first argument on the RHS, $Perf(R)$ is endowed with the structure of a sheaf of category over $B$ via the map $Spec(R) \to B$. And of course $B= *$ case recovers the absolute case.

I expect arguments of Antieau-Gepner about representability of moduli generalize to this relative setting, though I do not have a precise statement at the moment.

Footnote: As you do in the question statement I'm sweeping some presentability issues under the rug here to give the idea, $2QCoh(-)$ needs to be constructed carefully, but sounds like you know where to look for these constructions.