What is nonequilibrium thermodynamics?

An important topic in thermodynamics and statistical mechanics is the description of systems that are not in equilibrium. It is important because most systems are not in thermodynamic equilibrium and routinely exchange energy and matter with their surroundings. Somewhat surprisingly, the equilibrium thermodynamics of pioneers such as Boltzmann, Gibbs, and Carnot has sufficed for many years, in part, I think, because of its success at guiding the design of heat engines and describing chemical reactions. A theoretical description of nonequilibrium systems, though, still remains a challenge and active area of research.

So what is a nonequilibrium thermodynamic system? I am seeking an intuitive answer, not the unenlightened, yet all-too-common statement "a system that is not in equilibrium."

Unfortunately I cannot find the answer in any one place. I've read several research articles, particularly on active matter, which provide about zero insight to the question. This is probably because journal articles typically assume some familiarity with a topic. Wikipedia's page on nonequilibrium thermodynamics, which I linked to above, seems to provide a good answer in the form of a long description. However, I usually run into problems when trying to identify what about a particular system drives it out of equilibrium or why classical thermodynamics fails to describe the system. For example, on the Wikipedia page noted above under basic concepts, a system between two thermostats at different temperatures is described as a nonequilibrium system, even though basic heat engines from (equilibrium) thermodynamics texts are modeled in this way.

I suspect that a general definition of a nonequilibrium system is elusive because we usually must appeal to specialized statements about the system at hand, such as what spatial and temporal scales we are interested in, and whether the system is in a steady state.

My intuitive understandings about nonequilibrium steady states are given below:

1. Thermal, pressure, or particle density gradients are present, resulting in fluxes.
2. The behavior of the system changes with spatial and temporal scales.
3. There are many ways to drive a system out of equilibrium, so general descriptions must include the nature of the driving processes.
4. Macroscopic properties are not easily defined. Microscopic properties, however, appear easier to describe.

Addendum
The link about nonequilibrium steady states contains one important quality about these systems: work must continuously be performed on a system to maintain its state.