to shake off or shake out: to fling dust, dirt, or cloth free with a brisk shake
At its plainest this is a physical, everyday verb: to shake something out or shake it off with a brisk motion. In ordinary Greek of the period it was the word for beating dust out of wool or a cloak, knocking olives from a tree, or turning a house upside down in a search. The weight it carries in the New Testament comes not from the word itself but from the pointed gesture it names.
All four New Testament uses describe one deliberate act. Three shake the dust off the feet: the instruction to the traveling disciples in Matthew 10:14 and Mark 6:11, and Paul and Barnabas at Pisidian Antioch in Acts 13:51. The fourth shakes out the garment, Paul at Corinth in Acts 18:6. This is not a peevish flick of frustration but a formal, public sign, done before witnesses to mark a decisive break.
The gesture speaks the language of the Greek Old Testament, which a first-century audience would have caught. There God "shakes" the Egyptians into the sea (Exodus 14:27), and Nehemiah shakes out the fold of his robe as a self-binding oath that God will likewise shake out anyone who breaks his promise (Nehemiah 5:13). Against that backdrop the messenger who has delivered his warning and been refused shakes the dust away to disown the rejection, hand back the responsibility, and leave the dust behind as evidence to be produced on the day of judgment. Luke reports the same act with a close relative of this verb (ἀποτινάσσω) and, for the seventy, with a separate verb for wiping off (ἀπομάσσω).