I've found it impossible to find a definitive answer from mini themselves detailing the function of the sport button. The best i could find was a snippet from the dealers patrickmini describing the new features of the 2017 clubman, hope this helps.
"With the Driving Dynamics Control system, you can choose Sport, Mid, or Green driving modes. Red, blue, or green rings around the center display screen remind you which mode you’re in. Sport mode holds the transmission gears longer, and it stiffens the steering with some packages. It also enhances the response of the throttle. The Clubman always starts in Mid mode because it combines many of the best features of the Sport and Green modes. Green mode is the most fuel-efficient, and it has optimized fuel delivery, tweaked shift points, and a special coasting feature."