Gemara

(Mishnah): If a girl does Mi'un to her husband Reuven, he is permitted to her relatives, she is permitted to his relatives, and she is not disqualified from Kehunah;

If he gave her a Get and remarried her, and she did Mi'un, married someone else, and was widowed or divorced, she may return to Reuven;

If she did Mi'un and he remarried her and divorced her, and she married someone else, and was widowed or divorced, she may not return to Reuven.

The rule is, if the Get followed the Mi'un, she may not return to him. If the Mi'un followed the Get, she may return.

A girl did Mi'un, married someone else, and was divorced. She married someone else and did Mi'un. She married another man, and was divorced. The rule is, she may not return to one who divorced her, but she may return to one she did Mi'un to.

(Gemara): The Reisha shows that that Mi'un nullifies the effects of a Get.

Contradiction (Seifa): If a girl did Mi'un, she married someone else and was divorced... the rule is, she may return to one she did Mi'un to, but not to one who divorced her.

This shows that Mi'un to a man does not nullify another man's Get!

Answer #1 (Rav Yehudah): Different Tana'im taught the Reisha and the Seifa.

Answer #2 (Rava): We need not say this! Mi'un to a man nullifies his Get, but not another man's Get!

Question: Why should we distinguish?

Answer: Since she recognizes the gestures and winking of a man who divorced her, if he would be permitted to remarry her, he will entice her to do Mi'un to her new husband to remarry him.

Question: If so, Mi'un to a man should not nullify his Get!

Answer: She chose to leave him, so we are not concerned lest she heed his gestures.

Question: There is a contradiction about when she left the same man!

(Mishnah): If she did Mi'un to Reuven and he remarried her and divorced her, and she married David, and was widowed or divorced, she may not return to Reuven.

Inference: Had she left David through Mi'un, she could return to Reuven. This shows that Mi'un to a man nullifies another man's Get.

Contradiction (Seifa): A girl did Mi'un, married someone else, and was divorced. She married someone else and did Mi'un. She married another man, and was divorced. The rule is, she may not return to one who divorced her, but she may return to one she did Mi'un to.

Answer #1 (R. Elazar): Different Tana'im taught the Reisha and the Seifa.

Answer #2 (Ula): The case is, she was divorced three times. This makes her look like an adult, so we do not allow Mi'un to nullify a Get.

Question: Who are the Tana'im of our Mishnah (according to R. Elazar)?

Answer: We learn from the following episode:

Question: If Levi divorced a girl, and she married Moshe and did Mi'un, may she return to Levi?

Answer (Rav): A man was hired for 400 Zuz to ask this of R. Akiva in jail. He said that she is forbidden. R. Yehudah ben Beseirah said the same.

R. Yishmael b'Rebbi Yosi: They did not ask this. We permit (Eshes Ish, which when it is mid'Oraisa is) Chayavei Krisos through Mi'un, all the more so we permit Chayavei Lavin (Machazir Gerushaso)!

Version #1 (R. Yitzchak bar Ashi'an): Rav (who forbids a girl who did Mi'un to remarry a prior ex-husband who divorced her) agrees that she is permitted to her ex-husband's brother.

Question: This is obvious! The only concern was that she recognizes his winking. This does not apply to his brother!

Answer: One might have thought that we forbid the brother due to her ex-husband (lest she come to marry him). R. Yitzchak teaches that this is not so.

Version #2 (R. Yitzchak bar Ashi'an): Just like she may not remarry her ex-husband, she is forbidden to her ex-husband's brother.

Question: Why is she forbidden? She does not recognize his gestures!

Answer: We forbid the brother due to her ex-husband.

Rishonim

Rif and Rosh (13:4): The Halachah is, Mi'un to Reuven is Mevatel Reuven's Get (i.e. previously he had divorced her. He may remarry her even if she married and left someone else after Mi'un.) Mi'un to David is not Mevatel Reuven's Get (if Reuven divorced a girl and she married David and did Mi'un, Reuven may not remarry her). Since she recognizes Reuven's gestures and winking, if he would be permitted to remarry her, he might cause her to do Mi'un to her new husband in order to remarry him.

Nimukei Yosef (DH Masnisin): Mi'un after a Get proves that she is a minor, therefore it is Mevatel the Get and permits her to remarry him even if she marries someone else in between. 'The rule is' teaches that even if she married many times, she may return to anyone whom she left through Mi'un, but not one who divorced her. Even though a Diyuk from the Reisha suggests that Mi'un to David is Mevatel Reuven's Get, this does not override what the Seifa says explicitly, that it does not. Whenever a girl does Mi'un we are not concerned lest her husband entice her to leave another man. Presumably he tried to appease her to stay with him, and failed. Divorce is against her will; perhaps she still loves him. If he later regrets divorcing her, he may get her to do Mi'un to her new husband.

Rif and Rosh (ibid.): She is also forbidden to Reuven's brother. This is a decree due to the Isur to remarry Reuven

Nimukei Yosef (DH Gemara): David's Mi'un erased Reuven's divorce. Letter of the law, she should be permitted. However, if we did not decree against Reuven's brother, the decree against Reuven would not last.

Rambam (Hilchos Gerushin 11:16): If a girl does Mi'un to Reuven, it is as if she never married him. They are permitted to each other's relatives, and she is permitted to Kehunah. If she later married another man and was widowed or divorced or did Mi'un, she may remarry Reuven. Further, if Reuven divorced her and remarried her and she did Mi'un, and she remarried and was divorced, she may remarry Reuven. Whenever a girl does Mi'un to a man, even if he previously divorced her it is as if he never divorced her and she may remarry him.

Magid Mishneh: Likewise, in this case she is permitted to Reuven's relatives. Some say that even when she is forbidden to Reuven she is permitted to his relatives. All the more so when she is permitted to Reuven she is permitted to his relatives!

Rambam (ibid.): However, if Reuven divorced a girl and she married David and did Mi'un, Reuven may not remarry her, since she left him through a Get. All the more so he may not remarry her if she was widowed or divorced from David. She is also forbidden to Reuven's father, son and brother, like other divorcees, even though she left David through Mi'un.

Poskim

Shulchan Aruch (EH 6:2): If a minor did Mi'un, she is permitted to Kehunah. If she was divorced, she is forbidden. If she was divorced (from a marriage mid'Rabanan), remarried her husband and did Mi'un, she is permitted. If she was divorced, married a different man and did Mi'un, some permit her to a Kohen.

Tur: The Rambam says that if a girl does Mi'un to a man, even if he previously divorced her it is as if he never divorced her and she may marry a Kohen. It seems that the same applies if Reuven divorced her and she married David and did Mi'un. Even though Mi'un to David does not Mevatel Reuven's Get to permit her to Reuven, this is because we are concerned lest Reuven entice her to do Mi'un.

Beis Yosef (DH u'Mah she'Chosav v'Nir'eh): This is reasonable. However, perhaps Chachamim do not Mevatel a Get partially. Since Mi'un to David does not permit her to Reuven, it does not permit her to Kehunah.

Bach (DH u'Mah she'Chosav v'Nir'eh): The Beis Yosef agree that the Tur's opinion is primary. B'Di'eved, if she married a Kohen he may keep her.

Beis Shmuel (6): It seems that the Rambam is correct. We forbid her to Reuven's brother, even though there is no concern lest he entice her to do Mi'un.

Rebuttal (Shirei Korban Yerushalmi 72b DH Ein): The Rambam did not say that Mi'un does not permit another man's Get regarding Kehunah. Perhaps he agrees that it does! Also, there is a special concern if she would marry Reuven's brother. Since Reuven is often by his brother, he is prone to entice her. There is no such concern regarding Kohanim.

Shulchan Aruch (155:10): If a girl did Mi'un to Reuven, married another man and was widowed or divorced, she may remarry Reuven. Further, if Reuven divorced her and remarried her and she did Mi'un, and she remarried and was divorced, she may remarry Reuven. Whenever a girl does Mi'un to a man, even if he previously divorced her it is as if he never divorced her and she may remarry him. However, if Reuven divorced a girl and she married David and did Mi'un, Reuven may not remarry her, since she left him through a Get. All the more so he may not remarry her if she was widowed or divorced from David. She is also forbidden to Reuven's father, son and brother, like other divorcees, even though she left David through Mi'un.