Gemara

(Rav Yehudah, citing Rav Asi): If a child knows to discard a rock but keep a nut, he can acquire for himself, but not for others. If one can deposit an item by him and he will return it later, he can acquire for himself and for others.

(Shmuel): Both of these children have the same law.

(Rav Chisda): Shmuel means that either can acquire for himself, but not for others.

Question (Rav Chinena Vardan (against Rav Chisda) - Mishnah): To make Shituf Mavo'os (an Eruv to allow carrying in an alleyway on Shabbos), one designates a barrel of wine for all members of the Mavoy. Someone acquires on their behalf;

The following can acquire: the man's adult son or daughter, or his Yisrael male or female slave.

We cannot say that his female slave brought hairs (and became a Na'arah), for if so she is free, she is not his slave! Rather, she is still a minor, and the Mishnah says that she can acquire for others!

Rav Chinena was silent.

Answer (Rav Chisda): Rav Chinena should have answered that Shituf Mavo'os is mid'Rabanan. Chachamim were lenient to allow her to acquire, but mid'Oraisa, she cannot acquire.

Rav Chinena did not say so, for he holds that only Rabbinical enactments with an origin in Torah (e.g. to take the Lulav every day of Sukos) are patterned on Torah laws.

Bava Metzia 12a (Mishnah): Reuven receives Metzi'os found by his small (minor) sons and daughters, his male and female Kena'ani slaves, and his wife;

If his big (adult) sons or daughters, male or female (Yisraelim) slaves, or his ex-wife (even if he didn't pay her Kesuvah yet) found a Metziah, they keep it.

(Gemara - Shmuel) Question: Why did Chachamim say that Reuven receives Metzi'os found by his minor son?

Answer (Shmuel): When he finds it, he immediately brings it to his father. (He picked it up with intention to give it to him.)

12b: Shmuel argues with R. Yochanan:

(R. Yochanan): 'Small' and 'big' in the Mishnah do not refer to minor and adult;

Rather, any child fed by his father is called small. Any child who feeds himself is called big.

Rishonim

The Rif and Rosh (Eruvin 25a and 7:8) bring the Mishnah.

Rosh: R. Chananel says that according to R. Yochanan, who says regarding Metzi'os that an adult who is fed by his father is considered a minor, one cannot Mezakeh an Eruv through such a child, for his Yad (power of acquisition) is like his father's Yad. The Yad of his Eved Ivri is unlike his own Yad, even though he feeds his Eved. R. Yochanan discussed only a child, for normally he is always fed by his father. If Ploni feeds a Nochri (Hagahos R. Perla - i.e. a stranger), even for free, the Nochri's Metzi'os do not belong to Ploni. All the more so Ploni does not get the Metzi'os of his slave, who is fed because he works!

Rambam (Hilchos Eruvin 1:20): One can be Mezakeh food for an Eruv through his big children. He cannot be Mezakeh through his small children, because their hands are like his.

SMaK (cited in Hagahos Maimoniyos 10): R. Tam rules like R. Yochanan against Shmuel. The Rambam rules like Shmuel. The Yerushalmi brings only Shmuel's opinion.

Hagahos Maimoniyos (10): The SMaK could have cited the Bavli, which brings R. Yochanan's opinion only regarding Metziah, but not regarding Eruv! (It explains simply that big and small refers to adults and minors.)

Poskim

Shulchan Aruch (OC 366:10): One cannot be Mezakeh food for an Eruv through his minor children, even if he does not feed them, or through his Kena'ani slaves.

Gra (DH Afilu Im): Shmuel agrees with R. Yochanan regarding Eruv.

Shulchan Aruch (ibid.): He can be Mezakeh through his adult children, even if he feeds them.

Mishnah Berurah (57): Even though their Metzi'os go to their father, this is only an enactment to avoid resentment. They have their own Yad. If their father wants them to be Mezakeh for others, it works.

Magen Avraham (16): In a matter mid'Rabanan, a minor can be Mezakeh for others.

Shulchan Aruch (ibid.): Some say that he cannot be Mezakeh through his children if he feeds them, even if they are adults, or through his daughter before Bagrus, even if he does not feed her.

Beis Yosef (DH v'Chosav ha'Rosh): Tosfos and the Rambam say that a Na'arah's Metzi'os go to her father even if he does not feed her (lest he resent her, and marry her to a repulsive man).

Shulchan Aruch (ibid.): However, he can be Mezakeh through his minor son or his daughter after Bagrus if he does not feed them.

Rema: B'Di'eved, we rely on the lenient opinion in Eruvin. L'Chatchilah, one can be Mezakeh through any married adult child, even if he feeds him.