While cynically coincidental, my business is exclusively with the Sugya of the [temporary] impeachment of Rabban Gamliel in his Maariv Chova argument with R' Yahoshua (Brochos 27b).
אמרי עד כמה נצעריה וניזיל? בראש השנה אשתקד צעריה בבכורות במעשה דרבי צדוק צעריה הכא נמי צעריה תא ונעבריה
...How long will Rabban Gamliel continue afflicting R' Yehoshua? Last year on Rosh HaShana, he afflicted him; Regarding the firstborn, in the incident involving the question of Rabbi Tzadok, he afflicted him just as he did now, and forced him to remain standing as punishment for his failure to defend his differing opinion. Here too, he is afflicting him. Let us remove him from his position as Nasi. (And so they did).
While listing the procedure of electing the Sages, I don't recall a source for the Halachic considerations for the impeachment of the incumbent Nassi.
Does anyone deal with this question?