run, suddenly
驟 centers on sudden, rapid movement: a burst of running or something happening abruptly. The two given meanings are closely linked by the idea of a swift, unexpected dash.
驟 combines 馬 (horse) with 聚, which likely contributes the sound and the idea of gathering or suddenness. The character suggests a horse breaking into a sudden run or a swift gathering of speed.
A horse 馬 on the left and a gathering 聚 on the right: picture a herd of horses suddenly stampeding together, running all at once.
For シュウ, imagine a horse suddenly dashing off with a loud 'shoo!' sound: shoo -> シュウ, and the horse runs abruptly.