It depends on the country/state etc., but typically there is a process to "continue" a corporation from one country to another one.

Or one can often leave a corp hq in the old state and register the existing corp as a foreign corp in the new state.

US LLC's are a little more state-specific/diverse, so there'll be some details to look into on that as well.

As they say... speak to a lawyer! Good luck! ;)