Does $J = \sigma E $ hold when a charged mass is subjected to an external force? Let's say that I have a charged mass that is moved by some non-electromagnetic means such as gravity or fluid flow. 
I expect Maxwell's equations to hold in this case. However, I am confused as to how the relation 
$$J = \sigma E $$ 
will behave. 
I know that $\sigma$ is tensorial in general and that it can depend on a variety of factors (space, time, E, B...), however, I do not expect $J = \sigma E $ to hold linearly (in the context of Maxwell's equations) because the flow of current is not due to the electromagnetic field. 
I tried looking at this scenario in terms of the macroscopic formulation of Maxwell's equations as well (https://en.wikipedia.org/wiki/Maxwell%27s_equations#Macroscopic_formulation)  but all I could do was attempt to make mental analogies from this scenario to bound/free charges.
In some literature, I saw that $$J = \sigma E + J_s$$ (i.e. with some source term). This can make sense to me in the sense that the charged mass will contribute some specific $J_s$, making $J$ nonlinear. Otherwise, the other option is $J=0$.
Something is clearly amiss with my understanding, so I would appreciate it if someone could fix my perspective. 
My main concern is about how the J term in Maxwell’s equations (Ampere’s Law with Maxwell’s addition) and the equation for charge conservation (i.e. the continuity equation) will be affected (if at all) in this case. 
Thanks!
 A: $\vec{J}=\sigma\vec{E}$ is just the microscopic form of Ohm's Law. It only relates to conduction in certain types of material. It is by no means a general law of nature (and probably never should have been called a "Law").
In the materials and circumstances where Ohm's Law applies, charge transport due to gravity or other forces are not likely to be significant compared to transport due to electric field. 
However if they did, you are correct that Ohm's Law would not be a complete model for conduction in those cases.
From comments, 

Do you have suggestions for models that could serve better than Ohm's Law? Specifically for a general interpretation of J in Ampere’s Law (w/ Maxwell’s addition)?

Maxwell's equations tell us how currents produce magnetic fields. 
They don't tell us how magnetic or electric fields act on charge to produce currents.
For that you have to look at the Lorentz force law
$$\vec{F} = q\vec{E} + q\vec{v}\times\vec{B}$$
along with whatever other forces are acting on the charge. Then of course the current will depend on the mass of the particles carrying the charge, the number of density of charged particles present, etc.
When solving Maxwell's equations, the effect of the fields on the currents will normally be captured in the boundary conditions for the problem. If there is free charge in the same region as the fields are propagating, you are getting into the world of plasma physics, which I don't know much about.
