A well known model theory fact is that for any first-order theory $T$ the collection of universal consequences of $T$, written $T_\forall$, is a precise axiomatization of the class of substructures of models of $T$. This is related to the fact that a sentence is preserved under passing to substructures if and only if it is logically equivalent to a universal sentence.

Dually a sentence is preserved under passing to superstructures if and only if it is logically equivalent to an existential sentence. The characterization of models of $T_\exists$ is less clean, specifically it axiomatizes the class of elementary substructures of superstructures of models of $T$.

Suppose $\mathcal{L}$ is a two-sorted language with sorts $A$ and $B$. Given $\mathcal{L}$-structures $\mathfrak{M}$ and $\mathfrak{N}$, we'll say that $\mathfrak{N}$ is an *$A$-super-$B$-substructure of $\mathfrak{M}$* if $A(\mathfrak{N}) \supseteq A(\mathfrak{M})$ and $B(\mathfrak{N}) \subseteq B(\mathfrak{M})$ with the interpretations of all relation and function symbols agreeing on the common substructure $A(\mathfrak{M}) \cup B(\mathfrak{N})$. Intuitively speaking we've allowed $A$ to grow and $B$ to shrink when passing from $\mathfrak{M}$ to $\mathfrak{N}$. This relationship is obviously transitive.

We'll say that a sentence $\varphi$ is *preserved under passing to $A$-super-$B$-substructures* if for any $\mathcal{L}$-structures $\mathfrak{M}$ and $\mathfrak{N}$, if $\mathfrak{M}\models \varphi$ and $\mathfrak{N}$ is an $A$-super-$B$-substructure of $\mathfrak{M}$, then $\mathfrak{N}\models \varphi$. We may also requires that $\mathfrak{M}$ and $\mathfrak{N}$ be models of some particular theory.

A natural example of a sentence with this property is extensionality. Suppose that $B$ is a sort of sets of elements of $A$ and consider the extensionality axiom: $$(\forall x,y:B)(\exists z : A)((z\in x \leftrightarrow z \in y)\rightarrow x=y)$$

Adding more elements to $A$ cannot spoil extensionality, regardless of how you extend $\in$, and removing sets from $B$ cannot spoil extensionality.

This sentence has a particular syntactic form, specifically it is prenex and has the property that all $A$-quantifiers are $\exists$ and all $B$-quantifiers are $\forall$. Let's call the collection of sentences of this form $\mathcal{L}_{A\exists B\forall}$ (note that there is no restriction on the number of alternations). An easy argument shows that any sentence in this form is preserved under passing to $A$-super-$B$-substructures. My question is about the converse:

Question 1:Fix a theory $T$. Suppose that $\varphi$ is a sentence that is preserved under passing to $A$-super-$B$-substructures provided that both structures are models of $T$. Does it follow that $\varphi$ is equivalent to a sentence in $\mathcal{L}_{A\exists B\forall}$ modulo $T$?

Question 2:Let $T_{A\exists B\forall}$ be the set of consequences of a theory $T$ that are sentences in $\mathcal{L}_{A\exists B\forall}$. Are the models of $T_{A\exists B\forall}$ precisely the class of elementary substructures of $A$-super-$B$-substructures of models of $T$?