It just gets relocated to another part of the universe.
Would I be right in thinking that the agent that moves it to another part of the Universe is heat, flowing down a temperature gradient?
I'm not quite sure what you mean by that. Essentially in cooling one part of the universe, even if it is just one particle, the heat is transferred to another part of the universe, if it is just the particle next to the one being cooled. There are many different processes that can perform the exchange. In effect when one part of the universe is cooled another part is heated. As far as flowing is concerned, I think that is probably a classical concept, where heat "flows" from a high temperature source to a low temperature sink. When you are talking about temperatures near absolute zero there is no clear place for the heat to "flow" to because there is usually nothing around that is at a lower temperature. Getting those last few millionths of a degree requires the use of quantum effects and I'm not sure how to express that heat exchange.