How is Newton's first law compatible with transport equations? So, I'm studying non-equilibrium thermodynamics and the first thing I've learned is that the discipline has a first scope of generalizing the idea of transport equations.
Transport equations are generally of the form:
$$J=-A\cdot\frac{dX}{dx}$$
Where $J$ is the flux of something traveling across an area, $X$ is some quantity defined in each part of the physical system (viscosity, mass, temperature etc...) and $A$ is a proportionality constant. $\frac{dX}{dx}$ can be viewed as a gradient of that quantity $X$, and thus as a kind of force.
There are several transport equations related to different phenomena. For example, Fourier's law is a transport equation that relates the gradient in temperature with a heat flux, the gradient in temperature being a driving force of that energy flux between different parts of the system.

So, the conclusion here (at least what my teacher says) is that there can't be a flux of any quantity without a gradient of some kind (a driving force) and vice versa.
But this, even if it sounds reasonable, contradicts some ideas I had about Newton's first law of motion: if I have a solid rod moving across space in uniform linear motion then I can conclude that there is no force pushing it. The rod moves only because it has inertia and thus it will keep moving if no force is added to the mix. And if I imagine calculating the flux of mass inside a circular area as the rod traverses it then I can see there's some kind of constant flux of matter as the rod passes through it. So it seems that we have a situation where there's a flux but there is no driving force, only inertia. So it stops to be true that fluxes and currents arise from the unevenness of some quantity through space.
My question is where I've been misleading? Is it the fact that the rod I'm describing is not a thermodynamic system subject to an immense number of interactions, but can be viewed as a microscopic system (made of just one element) and thus transport equations are not relevant here? Is it that the flux of mass is not related to that particular force but to another generalized force I'm not talking about? Is it that the driving force as a generalized concept should not be confused with the actual mechanical force here for some reason? What am I missing exactly?
 A: I find it difficult to give in words an answer to your question.
For your example of a moving rod you have assumed that there are no forces in opposition to motion of the rod (eg friction) and so the "phenomenological constant" in your example is infinite.
Thus no pressure difference (force) is required to transport the mass in the same way that no potential difference is required to transport electrons (electric current) in a superconductor.
A: In order to have a (constant, non-zero) flux you need either no force at all or else a balance of forces. The situations ordinarily under discussion here are where there is either friction or a friction-like force, which acts to oppose motion (and thus flux). In this case you would need some other force, such as the result of a gradient, in order to overcome the friction. The coefficients are often named with names such as 'conductivity', which is a good name but the name might make the unwary forget the fact that it is a friction-like effect.
The generalized friction or resistivity is inversely proportional to the generalized conductivity. So if there is no friction, as in the case of the idealized situation discussed in Newton's first law, then the 'conductivity' is infinite.
In classical physics I think it would be correct to say that one 'always' needs some sort of gradient to have a flux, if one is saying that the idealization of no friction at all is just that: an idealization. For example, Newton's idea of a body moving in empty space is an idealization in the sense that space is never entirely empty (no vacuum pump will give a perfect vacuum). But in phenomena such as superconductivity and superfluidity you can have a persistent current which is never damped. And currents of this kind can also be found inside the electron structure of every atom.
A: The fluxes in your equation are supposed to be relative to an observer traveling with the mean velocity (see Transport Phenomena, Bird et al).  So for your rod moving at constant axial velocity v in the x-direction, the overall flux of heat is $$\Phi=\rho CvT-k\frac{dT}{dx}$$where C is the heat capacity of the rod.
A: You can, in case of the rod, find a frame of reference where no mass passes the area. There is no force pulling the particles through the surface. The concentration of particles in your rod is constant over the rod. The gradient is zero. There is no net motion in the rest frame of the rod.
If you look at Fick's laws, you can add the extra mass displacement induced by your whole system moving. In every frame there is a gradient which is not influenced by a global velocity. The gradient "force" is absolute. Present in every frame. If you compare the rest frame situations, for the rod and your concentration gradient system, you see there is nothing wrong.
A: TL; DR: microscopic vs. macroscopic motion
The difference here is akin to the difference between heat and work: one represents energy transfer on a microscopic level, whereas the other is the macroscopic motion of the same molecules as a whole. Another common microscopic/macroscopic pair is diffusion and convection, when we are talking about mass transfer. One could similarly find appropriate pairs for any of the transport phenomena in the table, given in the OP.
Microscopic motion does not break Newton's laws, but these laws have to be applied on the molecular level, taking into account the interaction between the molecules. The example of the moving rod, on the other hand, is a macroscopic motion of the rod molecules together, characterized by the motion of the center of mass.
Another fine point is that thermodynamic laws are usually derived and discussed in the system of reference where the body is neither moving nor rotating - they do not exclude the motion of the body (e.g., a gas reservoir) as a whole, which would occur in accordance with the Newton's laws.
A: 
So, the conclusion here (at least what my teacher says) is that there
can't be a flux of any quantity without a gradient of some kind (a
driving force) and vice versa.

This conclusion is correct and I do not see how that should interfere with Newton's laws. I would refrain from calling it a driving force because that might be an analogy but maybe not the best one.
Imagine in your thought experiment with the rod you put a large box around the problem with a divider in the middle. Your rod is on the left side, the right side is empty. There is a concentration gradient (1 and 0 rods) and when the rod travels through the divider you can assign a flux to this transport phenomena.
Now this flux will be discrete, but without changing anything but the number of particles you end up with the classical diffusion experiment. Each particle will not be accelerated by any force, just elastic collisions with the walls and other particles are enough to model the process.
