IPA: /ˈswɜːr.lɪŋ.li/
KK: /ˈswɜr.lɪŋ.li/
In a way that involves moving in circles or spirals.
The leaves danced swirlingly in the autumn breeze.
Swirlingly is formed from "swirl" (of uncertain origin, possibly imitative of the motion) and "-ingly" (a suffix used to form adverbs indicating manner). The word describes an action done in a swirling manner.
Imagine something moving in a circular motion — that's what swirlingly means, as it describes the manner of swirling.