From Han Yu's Tang essay 应科目时与人书: 忘一举手一投足之劳 - Han Yu asked a patron not to begrudge the slight effort of lifting a hand or moving a foot. The phrase then generalized to 'every little gesture'.
usage
Used to describe someone's bearing or style - usually 举手投足都 + adjective/verb, emphasizing that even tiny gestures show some quality (elegance, confidence, awkwardness). Pairs well with adjectives like 优雅,自然,从容.