The origin of the idiom wild-goose chase, meaning "a complicated or lengthy and usually fruitless pursuit or search," has nothing to do with the pursuit of the bird—although we can imagine that chasing after and catching one would have its difficulties. The original wild-goose chase was actually a game in which riders on horseback tried to follow and keep up with a lead rider on whatever course he set. The game's name derives from its resemblance to a flight of geese with a lead goose followed by others in formation.
The Wild-goose chase being started, in which the hindmost Horse is bound to follow the formost, and you having the leading, hold a hard hand of your Horse, and make hym gallop softly at great ease….
— Gervase Markham, A Discource of Horsmanshippe, 1593
Early figurative use of wild-goose chase appears in William Shakespeare's tragedy Romeo and Juliet during a scene in which Romeo and his friend Mercutio banter with each other. Urged by Romeo to keep up the repartee, Mercutio says, "Nay, if our wits run the wild-goose chase, I am done; for thou hast more of the wild goose in one of thy wits ... than I have in my whole five." This is said in response to Romeo's banter, "Swits and spurs, swits and spurs! or I'll cry a match" (swits means switches), which makes the allusion to the equestrian game clear. Mercutio uses wild-goose chase to admit he is not quick-witted enough to keep up with Romeo.
This "follow the leader" metaphor fades with the riding game. When the game became obsolete, people began to mistakenly assume that wild-goose chase referred literally to the futile chasing of the bird, leading to today's "fruitless pursuit" sense.
… led him on egotistical wild goose chases like investigating the validity of President Barack Obama's birth certificate.
— Ed Montini, The Arizona Republic, 23 Aug. 2018