Each ball becomes available once you reach certain Alleys. You can check this in the ball's description (tap on the ball).
When you reach the required Alley, you have a chance to get a ball of this Alley's cards by:
- Unlocking cases that you get for victories
- Completing missions
- Getting a Crown Case
- Taking a Free Item in the Shop (with a chance)
- Purchasing in the Shop
- Ranking in the top 15 of your League
When you receive a ball card for the first time, that ball is unlocked. You can get its cards from any case, including cases from lower-level Alleys.
Your League adds a permanent bonus to the number of cards you get from cases. The bonus is shown on the Leagues screen.
New balls do not drop often. If you've reached a certain Alley it doesn't guarantee that all balls of that Alley are being unlocked. The positive side of this is that the fewer balls you have, the easier it is to upgrade them.
But if your goal is to collect all balls, then you can purchase Collector's Cases in the Shop, which have a 100% chance of giving you a new ball.
Please note, that it's not necessary to purchase them. All balls, except premium and rank balls, can be unlocked for free during the game, but it might take some time.