Gemara

(Mishnah - R. Meir): If a Shogeg murderer used to have a position of authority, when he goes free (after the Kohen Gadol dies), he returns to it;

R. Yehudah says, he does not return to his old authority.

(Beraisa - R. Yehudah): "V'Shav El Mishpachto v'El Achuzas Avosav Yashuv" - (when an Eved Ivri (Yisrael slave) goes free,) he returns to his family, but he does not return to the authority of his fathers;

R. Meir says, he returns also to the authority of his fathers - "v'El Achuzas Avosav", like his fathers.

(Beraisa - R. Yehudah): "Yashuv ha'Rotze'ach El Eretz Achuzaso" - a murderer returns to his family, but he does not return to the authority of his fathers;

R. Meir says, he also returns to the authority of his fathers. We learn from a Gezerah Shavah "Shivah-Shivah" from a returning slave.

Yoma 12b (Beraisa - R. Meir): If a Kohen Gadol was unable to serve on Yom Kipur and another substituted for him, the first returns to his service afterwards. The latter has all laws of a Kohen Gadol;

R. Yosi says, the first returns to his service afterwards. The latter cannot serve like a Kohen Gadol, lest this cause enmity (with the first). He cannot serve like a regular Kohen, for we rise in Kedushah, but we do not decline.

(R. Yochanan): The Halachah follows R. Yosi.

Berachos 27b (Beraisa): On three occasions, R. Gamliel pained R. Yehoshua. Rabanan decided that they must choose a different Nasi. They picked R. Elazar ben Azaryah. After R. Gamliel appeased R. Yehoshua, they decided to reinstate him.'

(Rabanan): We cannot depose R. Elazar, for we ascend in Kedushah, but we do not descend! If R. Elazar and R. Gamliel will expound on alternate weeks, there will be jealousy! Rather, R. Gamliel will expound three weeks, and R. Elazar on the fourth.

Rishonim

Rambam (Hilchos Rotze'ach 7:14): Even though a murderer gets Kaparah (after the Kohen Gadol died), he never returns to his former authority. He is forever demoted from his greatness because this great failure came through him.

Ritva (Makos 13a DH Mai): The Mishnah discusses returning to one's own authority. It teaches the extremity of R. Yehudah, that he holds that he does not return even to his own authority, for the Halachah follows him. The Beraisa teaches the extremity of R. Meir, that he returns even to his father's authority. All agree that we do not give him a new authority that neither he or his fathers had. Perhaps R. Yehudah is stringent only for a murderer, but one who did a different Aveirah and repented can receive even a new authority, and all the more so he returns to what he or his fathers had.

Ha'Makneh (Kidushin 21b DH Ein): The Mechilta says that an Eved Kohen cannot be Nirtza (have his ear pierced), and he cannot be sold. If he cannot be sold, surely he cannot become a Nirtza! Rather, this is like R. Meir. He says that a Kohen cannot sell himself, for that Parshah mentions "l"Eker", to serve idolatry itself. If so, the Kohen could not return to his service (in the Mikdash)! However, Beis Din can sell a Kohen.

Tosfos (Yoma 13a DH Halachah): Why did R. Yochanan need to rule like R. Yosi? This Halachah will not apply until Mashi'ach! Rather, this pertains to a leader who was replaced due to Ones, The first returns to his position, and we also show honor to the second, like when R. Gamliel returned to be the Nasi.

Question (Maharsha): Also R. Meir holds that the first returns to his position!

Answer #1 (Hora'as Sha'ah): R. Meir holds that both of them remain Kohen Gadol, even though there is enmity, i.e. the first thinks 'they still let me serve only because I was anointed permanently. If not, I would be permanently deposed.' If a leader was replaced, R. Meir holds that we do not demote the latter at all, even though there is enmity. R. Yosi is concerned for enmity, therefore, the first returns to his position, and we show honor to the latter.

Answer #2 (Yishuv ha'Da'as): R. Meir holds that both of them are equal, but the first does Avodos that both of them cannot do. Regarding a leader, both of them serve, without concern for enmity. The Halachah follows R. Yosi, who is concerned for enmity, so we do not appoint the latter at all. If R. Elazar and R. Gamliel would expound on alternate weeks, there would be jealousy! The new Kohen Gadol does not continue to serve at all. There they enacted that R. Gamliel expound three weeks, and R. Elazar the fourth week, for there, there is not so much enmity.

Mordechai (Bava Basra 533): Ploni used to do a certain Mitzvah, e.g. Glilah (tying the Sefer Torah), and due to Ones or poverty the Tzibur gave the Mitzvah to Almoni. Later, Ploni became rich, and he wants to return to do the Mitzvah. Rabbeinu Meir answered that if when they gave it to Almoni, Ploni could have paid what he used to every year, but he did not want to, he waived his rights and pardoned it to Almoni. It suffices to have witnesses that Ploni agreed, or witnesses that Almoni became Muchzak in the Mitzvah and Ploni did not protest. This is a Chazakah with a proper claim (Ploni pardoned). However, if when they gave it to Almoni, due to Ones Ploni could not pay what he used to, and now he can and he wants to resume doing the Mitzvah and giving like he used to, he resumes. This is like a Kohen Gadol who was Tamei on Yom Kipur and another substituted for him. The first returns to his service after he becomes Tahor. The Gemara rules like R. Yosi. The Ri said that this is relevant to a leader who was replaced due to Ones, The first returns to his position when the Ones passes. He is no worse than a murderer, who returns to his former authority according to R. Meir. Even R. Yosi argues only due to a Gezerah Shavah from Eved Ivri. There, he does not return because he sinned, i.e he killed b'Shogeg, or an Eved Ivri sold himself or was sold for stealing. Here, he did not sin, so he returns.

Poskim

Shulchan Aruch (OC 153:22): If Ploni used to do a certain Mitzvah, e.g. Glilah, and due to Ones or poverty the Tzibur gave the Mitzvah to Almoni, and later Ploni became rich, and he wants to resume the Mitzvah, if when they gave it to Almoni, Ploni could have paid what he used to every year, but he did not want to, and he agreed with the rest of the Tzibur to give it to Almoni, he lost his rights. However, if when they gave it to Almoni, Ploni could not pay what he used to, and now he is able to and he wants to resume doing the Mitzvah and giving like he used to, he returns to his Mitzvah.

Magen Avraham (49): Mayim Amukim says that we do not change to someone else, even from an ignoramus to a Chacham, unless there is a reason to change. Even so, if there is a reason not to change, we do not change. If there is a clear proof that he despaired from the Mitzvah, he lost his Chazakah. Likewise, if a leader lost his position due to Ones, and the Ones passed, he returns to to his position (like Tosfos says). Tosfos is difficult. Also R. Meir holds that the first returns to his position! Perhaps he learns that the latter gets no honor from the position he temporarily fulfilled, lest this cause enmity. This requires investigation. The Mordechai connotes that if the first was deposed for any Aveirah, even b'Shogeg, he does not return to his authority, just like a murderer. The Rambam rules like this.

Machatzis ha'Shekel (DH vEfshar): It is difficult to say that Tosfos answered that we rule like R. Yosi to teach that we do not show honor to the latter. Tosfos does not connote like this. Also, we do honor the latter, like the case of R. Elazar! We must say that since they greatly favored R. Gamliel (he expounded three weeks out of four), there would be no jealousy. Tosfos holds that we honor the latter, but less than the first, to avoid enmity. However, it seems that Tosfos' Chidush is that we honor the latter. The Mordechai holds that we expound the reason why a murderer does not return to his authority (because he sinned), and the same applies to anyone who sinned.

Eshel Avraham (49): Rashi and the Bartenura rule like R. Meir. Chemed Moshe says that R. Elazar was not totally demoted because R. Gamliel was not deposed due to Ones, rather, due to a quarrel.

Gra (52): Pardon does not require a Kinyan. If the latter did the Mitzvah in front of the former and he was silent, he pardoned, like it says in Bava Basra (60a) about blocking a window.

Mishnah Berurah (112): We discuss when Ploni became Muchzak with the Tzibur's permission. The Radvaz (4:11) says that if he was Muchzak a long time and they were silent, this is a Chazakah.

Mishnah Berurah (115): The Poskim say that if a leader was deposed and reinstated, the replacement does not keep any share of the position, but we give him an honorable appointment for a need of the Tzibur. If the first was deposed due to an Aveirah, he does not return. Eliyahu Rabah was unsure if this applies only to a murderer because a calamity came through him.

Kaf ha'Chayim (178): One should not fight over any Mitzvah.

Kaf ha'Chayim (180): If a Rebbi needed to move to another city, and they appointed a new Rebbi, and the first returned, and a Rebbi died, perhaps he has no precedence for the position. However, since he left due to Ones, it is proper to give him precedence.