To "shake off dust from one's feet against a city or person" implied a solemn refusal to take anything away, even the very dust of their ground, but to leave it to witness against them (Mark 6:11); shaking off all connection with them, and all responsibility for their guilt and consequent punishment for rejecting the gospel.
These dictionary topics are from Fausset Bible Dictionary, 1949. Public Domain.
Fausset, Andrew R. "Entry for 'Dust'". "Fausset Bible Dictionary".