From Generation VII onward, usually offspring will have the same Poké Ball as the female Pokémon, or the non-Ditto Pokémon when breeding with Ditto. However, if a male and female Pokémon of the same species are bred (regardless of form), the offspring has an equal chance of inheriting either parent's Poké Ball.
So the female Pokemon will pass it down when bred with breed with anything but the same species. Otherwise it is an equal chance.
It should be noted that Master Balls, Cherish Balls, and Strange Balls cannot be inherited by any means. So yes, you can pass Sport Balls down through any one of the methods listed above.
Source: https://m.bulbapedia.bulbagarden.net/wiki/Pokémon_breeding
