First, of all, while there is a quantity called "heat" (the average kinetic energy of the air) there is no quantity called "cold". "Cold" is simply the absence of "heat". That is, the only thing that could move from one place to another is "heat".
More specifically, the faster moving molecules of air from the warmer room move into the colder moving room faster than the slower moving molecules of air move from the colder room to the warmer. So again, it is the "heat" that is moving.
(This reminds me of the old joke, "Which is faster, heat or cold?"
Answer: "Heat, because it is easy to catch cold!")