You said:

Ok, it was the "accumulation" part which was causing me to wonder,
Snip:
In other words if you had some sphere which was made of glass which was transparent to infrared radiation which is leaving the sphere but not transparent to any incoming infrared or heat from conduction from outside [or heat conducted from inside]. And the surrounding air was cold and low pressure, and the air inside the sphere was also low pressure and cold but say 10 C warmer than outside air- how long would it take to lower the temperature by say 5 C?

That is an interesting concept so lets investigate this thought experiment a little further.

For one, we must understand how a molecule can gain and loose kinetic energy.

As I understand it, there are basically two ways that the temperature [kinetic energy] of a molecule can increase, one might be considered as a collision with molecules that have a higher kinetic energy.
In the opposite function of loosing energy, such a molecule can collide with molecules that have lower kinetic energy and that will transfer some of the higher kinetic energy to the molecule with the lower kinetic energy.
If the collisions were the only method for the energies to interact, then under your specified conditions, the molecules would never decrease in kinetic energy, because the energy must always be conserved.

There is the other way that the molecule can increase in kinetic energy, and that is that it can absorbs electromagnetic energy that is of a certain frequency.
Conversely such an effect has its opposite which is the emission of electromagnetic energy. [photons] It seems that all types of molecules can absorb and emit infrared photons.

So, under your specified conditions, the IR energy would be lost and therefore the kinetic energy also would be lost, then in theory eventually the molecules would have no kinetic energy.

Now consider the interactions between the molecules if some of the molecules were radioactive, what would happen to the Alpha. Beta, & Gamma rays?
And what would happen to the kinetic energy of the molecules?