The short answer: It varies a lot depending on what you want the dart to do. The stuff in a tranquilizer dart is generally a surgical anesthetic (or mix) that works the same way as Heroin, if you give too much the animal overdoses and can die. read more
But tranquilizer darts are usually fired into thick muscle tissue, and then it takes a few minutes for the tranquilizer to get into the bloodstream and from there to the brain, in quantities large enough to be effective. Also the higher the dose the quicker it works, but if the dose is too high there is a risk of killing the animal. read more
No. Outside of TV and movies tranquilizer darts take a few minutes to take effect. Some anesthetics can take effect in as little as 40s, but those wear off just as quickly unless you give a continuous dose, so they aren't used in darts. read more