Computational equivalences between a calculus and an automaton model
As I said, I am not sure how to answer your question.

However, given the current rephrasing of it, it finally rung a memory
bell. I think that, starting sometime in the 1970's, people have been
studying what they called program schemes.

A set of slides (in French) I found on the web seems to give a
historical perspective on the subject.

Slide 14 is about Iterative program schemes, corresponding to Regular
sets. They correspond to computational structures of the form X=f(X,Y) and Y=a, which are only a functional version of iteration loops.

Slide 15 is about Recursive program schemes, corresponding to
Algebraic (also known as Context-Free) languages. They correspond to more complex recursive computation structure, using a stack memory as in early recursive programming langages.

I am not certain that is the kind of correspondance you are looking
for, as it correspond (as much as I remember) to the computing power of uninterpreted programs, i.e. the structural aspects of the computationq. But it may be worth reading.

There are many more references, including fairly recent ones, if you
serach the web with: "program schemes".