Green’s Dictionary of Slang

dust v.2

[all reflect an image of the dust raised by one’s speedy movement]

1. [mid-17C; mid-19C+] (orig. US, also dust along, ...it on, ...off, ...out) to rush off, to leave fast.

2. [late 19C+] (US) to overtake, to pass on the road.

3. [1930s+] (US, also dust off) to get rid of, to jilt.

4. [1950s] to pursue.

In phrases

dust one’s broom (v.) (also dust one’s bed, ... one’s blues)

[1930s–50s] (US black) to leave.

get up and dust (v.)

[mid-19C–1910s] (US) to act energetically; to run off quickly.