Difference between action-at-a-distance and a field according to Maxwell? 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.
 A: I cannot answer to what Maxwell knew and what he didn't. What I can tell you is what convinced me that the fields were real and the actions at a distance formulation is wrong. What convinced me was that action at a distance is not consistent with the ideas that momentum and energy are conserved. See, if you accelerate a charge, you can show that it produces electromagnetic waves, draining energy and momentum from that charge. Some of that energy and momentum can later show up in another charge responding to that wave. Without the fields existing as a way to store that energy and momentum, it would just vanish and reappear. With the fields, though, we can keep track of where the energy and momentum are at all times using the Maxwell stress tensor.
Feynman also gives an example of something that is a paradox only if you don't recognize that the electromagnetic field can carry angular momentum. See: The Feynman Lectures on Physics, Vol. II, Section 17-4 titled, "A paradox". In it, Feynman describes a situation where turning off a battery driving a solenoid will cause an isolated disk that carries the solenoid to rotate. The paradox is in asking where the angular momentum came from in the absence of any outside torques?
A: how did Maxwell know he was on the right track
Maxwell didn't even claim that he was on the right track (and the German weren't). In his treatise, he says that both the methods are valid, both the methods explained every electrodynamic phenomenon of his time.
The Maxwellians (Heaveside, Hertz, etc) were the first ones who thought Maxwell was on the right track. 
A: Assume you mean the Weber action or Schwarzschild-Tetrode-Fokker action. Wheeler-Feynman absorber theory applied the Schwarzschild-Tetrode-Fokker action. These action-ad-a-distance support advanced wave. 
According to the absorber theory a emitter sends a retarded wave in the same time also send a advanced wave.
In the John G. Cramer' transaction interpretation of QM, for each retarded wave which is offer wave, there is an advanced wave which is the confirmed wave.
Recently there is a new theory "mutual energy principle" in which the emitter sends a retarded wave and absorber sends a advanced wave. The two wave  must synchronized. See my publication: http://www.openscienceonline.com/journal/archive2?journalId=726&paperId=4042
The key point for "action-at-a-distance" is it needs a advanced wave. A emitter sends a retarded wave to absrober, then the absorber sends an advanced wave to the emitter, the time spends together is t+(-t)=0. Hence even action needs a time, but a reaction is immediately. This is the reason why a force can be non-local. 
