Gemara

42a (Rava bar Rav Huna): "One Nasi (for each tribe will acquire a share of Eretz Yisrael for his Shevet)" teaches that Zachin l'Adam she'Lo b'Fanav.

Objection: This cannot be! The division of Eretz Yisrael was not beneficial for everyone. Some prefer the valley, and others prefer the mountain!

Correction (Rava bar Rav Huna): "One Nasi" teaches that if orphans come to divide their inheritance, Beis Din appoints an overseer for them for their benefit and (even if this might entail) Chov (something detrimental).

78a (Beraisa - R. Shimon): A girl who converted before three years old is permitted to Kehunah. "Keep all the young girls (captives from Midyan less than three years old) alive for yourselves" applied also to Pinchas (a Kohen).

Sanhedrin 68b (Rabah): A minor cannot father children. It says "v'Im Ein la'Ish Go'el";

Question: Is there a Yisrael who does not have a Go'el (a relative who inherits him and is commanded to redeem him from slavery)?

Answer: The verse refers to a convert. It teaches that one must check whether an Ish (adult) has a Go'el (children), but one need not check a minor.

Kesuvos 11a (Rav Huna): If a minor wants to convert, we immerse him (in a Mikveh to convert him) on the volition of Beis Din.

Question: Does Rav Huna teach that Zachin l'Adam she'Lo b'Fanav (we may do something beneficial for a person in his absence)? A Mishnah teaches this!

Answer: We might have thought that conversion is not Zechus for a Nochri, for he prefers immorality. Rav Huna teaches that this applies only to an adult, who has tasted immorality. A child has not, so conversion is Zechus for him.

Support (Mishnah): If a girl converted below the age of three...

Suggestion: She was converted on the volition of Beis Din!

Rejection: No, she converted with her father. Children are happy with what their father does.

(Rav Yosef): When a child convert grows up, he can renounce the conversion.

Bava Metzi'a 71b (Ravina): A Nochri cannot have a Shali'ach, but (some texts - mid'Rabanan), he has Zechiyah (a Yisrael can acquire for him). He is like a minor, who cannot have a Shali'ach, but he has Zechiyah (mid'Rabanan)!

Rejection: No. A minor (Yisrael) has Zechiyah, for he can have a Shali'ach when he matures. A Nochri can never have a Shali'ach, so he does not have Zechiyah.

Rishonim

Rif, Rambam and Rosh (Yevamos 16b, Hilchos Isurei Bi'ah 13:7 and Kesuvos 1:23): We immerse a minor to convert him on the will of Beis Din. It is Zechus for him, and Zachin l'Adam she'Lo b'Fanav.

Question (Tosfos Kesuvos 11a DH Matbilin): Zechiyah works through Shelichus. I.e. since it is good for him, we know that he wants the Shali'ach to acquire for him. A minor has no Shelichus (Bava Metzi'a 71b). How can he have Zechiyah? Also, before conversion he is a Nochri. We said that a Nochri has no Zechiyah even mid'Rabanan, for he will never have Shelichus!

Answer #1 (Tosfos): Here also he has Zechiyah mid'Rabanan, because in this case (he converts so) he will have Shelichus. Alternatively, since he becomes a Yisrael through this Zechiyah, he is like a Yisrael regarding this Zechiyah.

Question: If his conversion is only mid'Rabanan, how can we permit him to marry a Bas Yisrael and say that his Kidushin takes effect?

Answer (Tosfos): Rav Huna holds like the opinion that Beis Din can uproot the Torah and permit a forbidden action (Yevamos 89b). The Gemara could have said that this is Rav Huna's Chidush.

Ritva (11a DH v'Yesh she'Pirshu): If conversion of a minor (Levi) is mid'Rabanan, if he was Mekadesh Leah and then David was Mekadesh her, she needs a Get from David. We cannot rely on Levi for any Torah law, e.g. Shechitah. Rashi explains that if Levi retracts, retroactively he was always a Nochri. A female minor convert may marry, for if she will retract the Isur Bi'ah was only mid'Rabanan.

Note: Levi cannot be Mekadesh until he matures. After that he cannot retract! The Ran (4a DH Garsinan) allows a convert to protest even after maturing, until he follows a Torah law. The Halachah follows Rav Huna who says that we immerse a minor on the volition of Beis Din. It is not a Mitzvah for Beis Din to circumcise Nochri children. Rather, if he came or his mother brought him or Beis Din happened to do so themselves, it takes effect.

Question: Sanhedrin 68b connotes that there can be a minor convert mid'Oraisa!

Answer (Tosfos): If a pregnant Nochris converted, her son is a convert (Yevamos 78a).

Rebuttal (of Answer #1 - Ketzos ha'Choshen CM 243:7): The Gemara in Kidushin (42a) teaches there is Zechiyah for a minor when it is pure Zechus.

Answer #2 (Tosfos): Some texts in Bava Metzi'a say Stam that a minor has Zechiyah. It is mid'Oraisa, even for a Nochri who comes to convert.

Question: Zechiyah works through Shelichus. A minor has no Shelichus!

Answer: He has no Shelichus for something that is partially Chov. Conversion is totally advantageous, so he has Shelichus for it.

Ritva (ibid.): If the conversion is mid'Oraisa, we rely on it for Torah laws. Since he grew up like a Yisrael, we are not concerned lest he retract. Some learn that it is mid'Oraisa from Kidushin 78a. This was not used to support Rav Huna, for perhaps Yisre'elim raised the captives and converted them before three. Also the Rambam (some texts - Ramban) holds that the conversion is mid'Oraisa.

Note: There were 32,000 girls below three. Surely, many were almost three, and were raised by Yisrael for a very short time before converting.

Answer #3 (Tosfos Sanhedrin 68b DH Katan): Zechiyah of conversion is different from other Zechiyos. Beis Din does not acquire for him. The convert acquires himself. Our ancestors, including minors, converted at the time of Matan Torah.

Question: The Gemara said that the conversion is based on Zachin l'Adam she'Lo b'Fanav, and asked because it thought that there is Chov!

Answer (Tosfos): If there was Chov, it would work, but Beis Din would not do so!

Answer #4 (Ketzos ha'Choshen, ibid.): Beis Din is different (it can be Zocheh even when a person could not).

Poskim

Shulchan Aruch (YD 268:7): If a minor Nochri has a father, his father can convert him. If he has no father and he came to convert or his mother brought him, Beis Din converts him, for this is Zechus for him and Zachin l'Adam she'Lo b'Fanav.

Piskei Din Rabaniyim (10 p.194:7): According to the opinion that conversion of a minor is mid'Oraisa, perhaps the same applies to conversion of a Cheresh (deaf-mute), even though he will never be obligated to do Mitzvos.

Shulchan Aruch (ibid.): Whether a minor was converted by his father or by Beis Din, when he matures he can protest.

Chasam Sofer (YD 253 DH v'Gam): Many Rishonim in Shitah Mekubetzes say that a convert who converts with his parents cannot retract. The same applies whenever it is a big Zechus, e.g. the captives from Midyan who converted to stay alive. The Rif never mentions retracting. The Rambam and BaHaG hold that if a minor's parent brings him to convert he cannot retract, even if the parent did not convert. The Rosh could agree. In any case when the parent converts, one may bless on the Tevilah of the child. We treat him like a convert in every respect, but we do not give to him money before he matures. However, if he will retract, retroactively his touching and Bi'ah disqualified wine and a Bas Yisrael.

Rebuttal (Ritva ibid.) The girls did not need to convert. They could have been kept alive for slaves!