I'm not an expert, but from what I understand this effect is due to resistance. As I learned in high school physics, there are basically 4 things that effect an objects resistance:

1. The material (i.e. Copper conducts better than aluminum)
2. The length (i.e. a 10 foot cable is more resitive than a 2 foot cable)
3. The cross section (a thick cable is less resitive than a thin one)
4. The temperature (A hot circuit is more restive, hence cooling fans in computers)

So that would mean that if the LED in the move were to heat up, the system would become more resistive and there would be less voltage turned into light.

It is true.

I did a Chemistry experiment, where we poured Liquid Nitrogen on LEDs and they would get brighter (due to lack of resistance induced by temperature).

Putting LEDs in boiling water made them dimmer.

Obviously neither of those things will (hopefully) not happen to a Move Controller, but thats why it changes in brightness.

This article offers a good explanation of how LEDs function.

The Cliff's notes version is that "LEDs typically change in brightness and color with temperature. As the temperature rises, the output decreases and the color shifts towards the higher end of the spectrum."

Some of the background as to why it increases increases in brightness: "...There are several factors that can affect the [wavelength]. One of these is temperature. As the ambient temperature rises, so to does the LED wavelength. This increase will typically be from 0.1nm/°C-0.2nm/°C depending on the type of LED used."

As the temperature decreases, so does the LED wavelength, making the LED brighter.

Any diodes have negative temperature coof. Some around 2.1mV/°C.

Output light is a power. If voltage across the LED drops 2.1mV/°C and current is constant - power goes down, light intensity too.