The higher elevations of the western Dakotas are far more prone to early snowstorms than the eastern Dakotas and Minnesota. There are also more very warm days out west during fall. The reason for both of these this is not specifically the higher elevation, but the east-to-west upward slope of the land.
When a low-pressure system moves across the Northern Plains, it develops an east-to-west wind field along its northern side. Air goes from an elevation of around 1,000 feet above sea level to around 3,000 feet or more in the far west. This results in about a 10% drop in air pressure, which forces the air to cool according to Charles’ Law. This is also why air being released from a tire feels cool. In any case, this cooling can sometimes be enough to turn rain to snow. The reverse process can result in down slope winds from the Rockies bringing very warm air into the western Dakotas.