There are two primary ways to 'lose' citizenship in an alliance.
RENOUNCING CITIZENSHIP
With a single specific exception, gaining citizenship with a new alliance requires the individual to renounce their citizenship of their original (or current) alliance.
In order to renounce citizenship it must be done in person, at an embassy of the alliance being renounced. This is usually done while in the process of gaining citizenship in another alliance, and until the new citizenship process is completed the individual is considered 'stateless'. Being 'stateless' is not the same thing as being considered 'unregistered', though it does limit an individual's rights considerably.
The exception of course is the EAOS, where inductees (both permanent and temporary) gain automatic dual-citizenship for the duration of their induction into the EAOS, unless the individual specifically chooses to renounce their alliance of origin. Temporary inductees become denaturalised EAOS citizens at the end of their induction term.
NOTE: Renouncing citizenship of an alliance ONLY removes your legal protections and rights given to you by that alliance. The majority of the legal obligations an individual holds to their alliance of citizenship remain for some time even after renouncing their citizenship, such as taxes, contracts, and so on.
DENATURALISATION
As opposed to actively choosing to remove citizenship yourself, losing citizenship without your choice is far more difficult.
The most commonly known form is by 'going unreg' - specifically stealing alliance technology and resources and heading into deep space, never to return. This does not count as renouncing your citizenship, instead this process forces the alliance in question to 'denaturalise' the individual who 'goes unreg' and force-remove their citizenship due to their breaking of a specific clause enshrined in interalliance law: by 'going unreg' the individual has committed high treason in a fundamental, irreversable way.
The Grand Amnesty removes the crime of treason and other negative aspects of 'going unreg' for anyone who did so prior to 21st August 2304, but does not restore their citizenship. Individuals who have 'gone unreg' after 21st August 2304 are not covered by the Grand Amnesty unless this is specifically added in an amendment to the Grand Amnesty by the alliance government of origin. Otherwise the individual is considered a traitor and guilty of high treason, etc.