You get the option of moving to a new generation once one of your adults passes away AND they have previously had children. No children to take over the house, game over.
To quote LadyCFII: "If you tap the "Family" button on the main screen, you will be taken to the family tree, where you should see a button at the top labeled "New Generation." Tapping that button will take you to the screens where you can select the child who will inherit the house and bank account."
The main screen refers to the screen that is open when you are playing the game.
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Kathy
Lead Tester
Last Day of Work