The ability to move in-game came with a patch.
However, if you're like me and never even used it until doing a challenge file:
You must have at least 20k simoleans as a "deposit" to move. Through the smartphone, it's under the travel and real estate option, simply just Move. Then you have the option to either just move to another house in town, or move to a new world. The game will prompt you to save before moving. Once you've chosen your new residence, it'll prompt you to save again. One thing that happened during the challenge file is, apparently, self employment gets reset (in my challenge file, the eldest child was a self employed painter, and when they moved to the last world, his self employment reset to 1) - though this could have been a bug, too.
Good luck with your dynasty!