When an alstirsus dies the Hillenistic kirijksuses from all around Lethea will meet in Athe to discuss who the next leader of the faith should be. Since there are thousands of kirijksuses in Lethea and many of them are too busy to attend, multiple kirijks choose one kirijksus to represent them.
Grounds for Removal/Dismissal
Only the Goddess can dismiss an alstirsus, but because the alstirsus is the only one who can supposedly communicate with her, she could theoretically ignore the Goddesses demand and continue serving as alstirsus.
The same problem applies to those who want to resign from the job. Although normally illegal, the alstirsus could claim that the Goddess demanded her resignation, but doing such a thing would mean that the alstirsus' reputation would suffer greatly.