My question is more on a historical note that involves Maxwell’s equations. Besides the information that I have obtained from textbooks, I am mainly getting it from History of Maxwell's Equations and was not able to find or understand the question below.

I first started thinking about this last semester while tutoring a mechanics student who believed that action-at-a-distance was a correct way of thinking about gravitational interactions between two objects. I explained to this student that the “correct” interpretation was the idea of a “field.” And it was here that I stated asking myself how did Maxwell know he was on the right track in developing the field idea from Faraday’s work? From my readings, I believe that I understand that he first developed his mechanical analog with an imaginary weightless and incompressible fluid flowing (where the stream lines are the electric and magnetic field lines) through a porous medium to reach his conclusions.

It seems to me that at an electromagnetic boundary, the action-at-a-distance theory would not be able to account for boundary conditions (in post Maxwellian language) established by a field. Even though I typically think of boundary conditions when referring to electromagnetic wave propagation across a boundary, I feel that somehow Maxwell “knew” this before he developed the electromagnetic wave equation and its associated boundary conditions:

$$ϵ_1E^{\bot}_1=ϵ_2E^{\bot}_2; {\;} {\;} E^{\parallel}_1=E^{\parallel}_2;{\;} {\;} B^{\bot}_1=B^{\bot}_2; {\;} {\;} \frac{B^{\parallel}_1}{{\mu}_1}=\frac{B^{\parallel}_2}{{\mu}_2}$$

I would appreciate someone giving me insight on this historical question or setting me straight on were I got it wrong.