Let's label a first order theory $T_1$ as infinitely equivalently descending if and only if there is a sequence of equi-interpretable effectively axiomatized theories $T_1 \supsetneq T_2\supsetneq T_3\supsetneq ...$
Where $T_i \supsetneq T_{i+1}$ means that every theorem of $T_{i+1}$ is a theorem of $T_i$, but not the converse.
Is it the case that every theory that meet Godel's incompleteness theorems, like PA and ZFC, etc.. are infinitely equivalently descending?
Basically every interesting theory has this property.
Let $S_i$ be the theory saying "Either there are at most $i$ many elements of the universe, or the universe is infinite."
(More precisely: for each $n$, let $\varphi_n$ be the sentence "There are at least $n$ many elements in the universe" and let $\psi_n$ be the sentence "There are at most $n$ many elements in the universe." Then $S_i=\{\psi_i\vee \varphi_j: j\in\mathbb{N}\}$.)
Clearly $S_1\supsetneq S_2\supsetneq S_3\supsetneq ...$ (any structure with exactly $k$ elements is a model of $S_k$ but not of $S_{k-1}$), and any theory with no finite models satisfies $S_1$. So:
When we restrict attention to finite languages, we can strengthen this substantially:
For suppose otherwise. Each finite structure $\mathcal{A}$ is characterized up to isomorphism by a single sentence $\theta_\mathcal{A}$. Suppose $\mathbb{A}=(\mathcal{A}_i)_{i\in\mathbb{N}}$ is a sequence of distinct finite structures, each not satisfying $S$. Then for each $k$, let $$R_k=\{\alpha\vee(\bigvee_{1\le i\le k}\theta_\mathcal{A}): \alpha\in S\}.$$ The models of $R_k$ are exactly those of $T$ together with $\mathcal{A}_1$ through $\mathcal{A}_k$. As above we have $$S\supsetneq R_1\supsetneq R_2\supsetneq ...$$
So - as long as our language is finite - the only theories which don't have your property are those which hold of all but finitely many finite structures. And very few of those theories are actually of interest.
(And it's even better - since checking satisfaction in a finite structure is decidable, we can find an appropriate $\mathbb{A}$ effectively in $S$ if such exists in the first place. So any computably axiomatizable theory in a finite language which fails in infinitely many finite structures actually bounds a computable descending chain of proper computably axiomatizable subtheories.)