To move cities in BitLife, you will need to use the Emigrate option that is located under Activities. Unfortunately, this is the only real way to get out of town, and it means leaving the country you are currently in.
There is no option just to move to a different city. Because of this, you will need to just pick one of the countries that are available and see if they will take you in.
If you don't like the set you get, you can just close the prompt and open it back up to get new options.The odd part about this is obviously you might just want to leave the city you are in and not the country, but that option isn't available to you.
Emigration is usually used to get out of a country you've committed major crimes in, so you can keep a low profile somewhere else.
You will have to worry about the country being willing to take you in, you will also have to pay a fee of some sort to make the jump, so hopefully you've got enough cash available to make the payment.
If you are wondering if you get to pick the city you land in, well, you don't get to choose! It seems to either pick the capital city, or whatever the major city of the country happens to be.
If you emigrate back to the country you were previously living in, it looks like you end up back in the same city you left from.
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