When light moves through an object and then slows, where does the lost momentum of the light goes?
Light actually never slows down. If it slows down, then it won't be light anymore. It is only the average over absorption and re-emission of photons in atoms that makes it look like that the light is travelling slower.
The momentum of photons is transferred to the atoms which absorb it and is again recovered when they re-emit them.
To understand this, you need to have a general idea of how the QED theory works.
That being said, we can only ever observe the effects of light, it is not something that can be caught, or observed directly. This is because it is more or less the vibration of charged particles.
QED theory explains that light is not a discrete instance that actually "travels". It is more of an energy transfer between charged particles via coulombic force. These charged particles are not mass less, so time is required for them to accelerate. The more charged particles in the path of "travel, the greater the collective acceleration time of light through a specific material.
This is why light will return to its original "speed", given that it is entering the same substance that it left when first changing materials.