Rules and regulations will differ from country to country, but provided you aren't a professional footballer (have a contract with your existing club) you are free to move to another league at will. Different federations will have some other restrictions (for example, I believe within most European leagues you cannot play for more than two clubs in a single season) but generically movement of amateur players is unrestricted. A transfer request needs to be submitted for professional players, or players moving between clubs within the same federation.
In Australia, and likely other countries with reputable football federations, you will be required to undergo an international clearance. This will be requested by the club you join in your new country and will be processed by their football association. They will liaise with the federation the player last played in (the parent federation) to determine whether the player is under any form of suspension or ban. No action is required by the player themselves, other than to inform their new club of the last club they played for.
When I have had to submit these forms, the usual response time from submission to clearance for play was 2-5 working days. If a request elapsed 3 weeks without a response from the parent federation, as can be the case with less organised federations from developing countries, the clearance is usually granted as de facto.