Why the games stop
Each game has a daily time limit, set to 10 minutes to begin with. You can change it to 5, 15, 20 or 30 in the grown-ups' page, which is behind a two-second hold on the little gear. 10 seemed like the right balance to give my son two games a day. The time starts again each new day.
When the time is up, the game doesn't cut off in the middle of something. It finishes the round, then the character gets sleepy and says it's time to stop.
By default every game keeps its own time, so a child who has used up one game can still play another. If you'd rather have one limit for everything, switch on "Share one daily limit" in the grown-ups' page of any game, and every game on that device counts towards the same time. The time is only kept on your own phone or tablet, and nothing is sent anywhere.