These words are synonyms when they mean to move something rapidly back and forth or in circles. Brandish means to wave or shake a weapon, especially in public, so as to threaten or intimidate someone: The drunk at the bar suddenly brandished a gun, and everybody scrambled for cover. Flourish implies a display of self-confidence, triumph or merely high-spirits: The victorious army flourished a captured flag.

Swing , in this sense, means to move rhythmically to and fro or backwards and forwards, like the pendulum of a clock or the clapper of a bell. It may imply a wide, sweeping motion, even rotation: to swing an axe; Huckleberry Finn recommended swinging a dead cat over one’s head to cure warts. Wield emphasizes the command one has over the sue of a weapon or instrument: to wield a hammer and chisel; to wield forceps with consummate skill. Sometimes wield approaches brandish in meaning: He turned to towards me and suddenly wielded a short, ugly knife.