, , and are often used in Japanese as if they had their own volition, particularly in modifying clauses.
You can say that a person's is walking off into the distance () -- this really means **that person** is walking into the distance, and the perspective of the scene is (the speaker) watching their back as they go.
So in this case, the speaker is following their (i.e. behind them) as it (as they) ``, `advance in a manner of kicking the piled-up snow about`.