The assumption made in the fluid models depends on the relation between plasma properties and local variables.
For the sake of simplicity let's focus on electrons, even if the concepts can be extended to ions.
Local variables are, for example, gas temperature, density, composition, electric field and so on.
Plasma properties are electron energy or temperature, the rates of electron induced processes, electron mobility and diffusion, and so on.
For each property of the plasma you need a function relating the plasma parameters to the local variables. The way you define this function determines the level of approximations.
A common approach is the local field approximation, where the stationary eedf is calculated as a function of the local variables. From the eedf all the plasma properties can be derived. See for example the attached paper. In this case neither local equilibrium nor maxwellian distributions are considered.