The first challenger is simply called the Challenger. All other challengers are in Standby. A Challenger attempts to beat all 13 leaders and become the Champion. If the Challenger fails, that Challenger is bumped all the way down to the back of Standby. If the Challenger defeats the Champion, the Challenger becomes the new Champion, and the former Champion is bumped down to Elite Four 4. The previous Elite Four 4 is bumped down to Elite Four 3, and so on. This diagram should explain it:

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If a leader wishes to resign, the leader under them takes the spot. The leader under them takes their spot, and so on. If and when Gym Leader 1 is bumped up to Gym Leader 2, the next Challenger has the choice to take the spot. If they do not, the choice goes through standby. If nobody takes the spot, any leader can fill in for them.