How does static equilibrium work in the body?
Static equilibrium works in a body by shifting the body from high energy states to low energy levels.
You can see this questions in two flavors: micro and macro. My answer is for the former, for the latter I cannot see an answer, "it just happens." See that the macro is explained from the micro, e.g. gas pressure can be explained, using Boltzmann they, by molecule speeds; heat can be explained by vibration of particles.
Static equilibrium is characterized by constant properties, e.g. temperature or pressure. Or even easier ones to grasp such as the simple fact that a body is not moving.
For example, you take a bar of metal, apply high temperature in the two extremes, by the current ideas about heat, the particles is going to vibrate more in the extremes then this vibration is going to propagate like an infection, soon, all the particles from the extreme to the center will no longer increase their vibration, notwithstanding they have different vibrations, high in the extreme and low in the center, this is static equilibrium, in spite of the lack of homogeneity in temperature, it is constant in time for a single point.
The same idea can be applied to a gas, using pressure as reference to static equilibrium. Please, do not misguide yourself, the fact that a body is in static equilibrium, does not mean that it is stopped, for instance, on osmosis, the concentrations in two sides can be constant, but particles still cross the membrane, but come back and so on. This situation could be better called "dynamic equilibrium."