Warning! this pushes the limits of every thermodynamic term and principle ever published.
Slowing down was the first part of the basic of basics of a thermodynamic system in equilibrium with steady state energy flow. It is not equilibrium or steady state, it is a combination where the equilibrium and steady state are conditional on each other, conditional equilibrium. Determining the energy flux that is steady state is the simple part.
The highest energy determines part of the conditional equilibrium, the energy level of the largest thermal mass the other part. If the lower energy portion has a very large thermal mass relative to the higher energy portion, the duration of the internal conditional equilibrium would be limited by the thermal mass of the warmer object. If the thermal mass of the two objects is equal and no energy is added to the system, the energy transfer from warm to cold would decrease to zero at the average energy of the two objects. This case for no energy added to the system would produce the longest duration of conditional equilibrium. If the thermal mass of both objects was infinite, there would be true equilibrium. If less than infinite, then the energy flowing from the box would decay to the average energy content, then decay more slowly as the internal objects inside are in "true" equilibrium. In a complex thermodynamic system there would never be a true equilibrium, only various conditional equilibria. If fact, without a phase change a large enough mass to undergo the phase change, this special cause of conditional equilibrium could not exist.
When a phase change is involved, the temperature of the phase change would be maintained until the phase transition is complete. The total energy required for a complete phase transition can be used to determine the slope of the decay to true equilibrium.
Example: If heat of fusion of water is 4000J/kgK and if the energy flow from the warmer to colder is equal to 4000J/kgK, then the latent heat of fusion of water is controlling the internal steady state. The temperature of the heat of fusion is 273.15K for fresh water, the temperature of the heat of fusion of salt water is 271.25K, in a salt water environment, the lower energy object effective radiant energy would vary between salt water freezing and fresher ice melting.
This is not your typical thermodynamic equilibrium. In a complex non-linear thermodynamic system there would be no true equilibrium. There would be a steady state maintained by the heat of fusion or by the heat of evaporation. If you place a large block of ice in a perfectly insulated box, the temperature of the box would remain at the melting point of ice until all the ice were melted. The steady state is conditional on the ability to maintain the phase change and the external equilibrium conditional interior maintaining a constant temperature.
Slowing Down Part III