Giving it to a Katan:
On the first day of Sukkos in Eretz Yisrael, and the first two days of Sukkos in the Diaspora, one is not to give his personal Daled Minim to any child below the age of 13 for a boy and 12 for a girl, to fulfill the Mitzvah. Thus one must purchase a separate set of Daled Minim for the children to shake. Alternatively, on the first day in Eretz Yisrael, and on the second day in the Diaspora, one may give the child his personal Daled Minim if all the adults have already fulfilled the Mitzvah. However on the first day in the Diaspora one is not to give children his personal Daled Minim even after all the adults have already fulfilled the Mitzvah, as one must use his own set of Lulav also the next day.
If the child does not have his own set: If one does not have another Lulav available to give the children then he may give the children his personal Lulav as a borrowed item, and not as a present. Thus he is not to say “a present on condition to return”, as he does when giving it to adults. Nevertheless some Poskim rule that the child may not say a blessing on this shaking.
If one gave the child the set as a present: If one did give the child the Daled Minim as a present then no other adults may fulfill their obligation with this Lulav until the start of Chol Hamoed. If one gave the child his Daled Minim without mentioning anything at all, and he also did not have anything in mind, then it is considered lent to the child and other adults may still fulfill their obligation with it even on the first days of Sukkos by having it given to them as a present.
E. Chol Hamoed:
During Chol Hamoed one does not need to use a Lulav which he owns. One may use another person’s Lulav without permission, assuming he doesn’t mind. However it is forbidden to remove the Lulav from its place unless the owner was asked. [Nevertheless it is proper to give another person one’s Lulav as a Matana Al Menas Lehachzir also during Chol Hamoed, and this is beneficial for both the giver and the receiver. So was the custom of the Rebbe.]
May a child give another child his Lulav and Esrog to shake?
Yes, if the child is above age 6.
May one buy Daled Minim from a child?
One may not buy any of the Daled Minim from a child under Bar Mitzvah unless the child does not own the Daled Minim and is selling them on behalf of another person. If one already shook Daled Minim that was purchased from a Katan then on the first day(s) one is to shake again without a blessing.
 Michaber ibid
The reason: As a child is unable to return a present.
 M”A 658/8; P”M 658 A”A 8; Bikurei Yaakov 658/18; Chayeh Adam 152/11; Igros Moshe 3/95 See Ketzos Hachoshen 241/4; Chanoch Lanaar 25
 Michaber ibid
 M”B 658/23; Biur Halacha 649 “Pesulei”
 Opinions brought in M”B 658/28; Ashel Avraham Butchach Tinyana
The reason: As one fulfills the Mitzvah of Chinuch even if the child does not fulfill the Biblical Miztvah.
Other opinions: Some Poskim rule one does not fulfill any Chhinuch obligation if the child does not own the set of Lulav. [M”A 658/8; P”M 658 A”A 8; Bikurei Yaakov 658/18; Chayeh Adam 152/11]
 Chayeh Adam 152/11
 Michaber ibid; Biur Halacha “Eino Muchzar”
 Chochmas Shlomo 658
 Hayom Yom 15th Tishreiy. So is implied from the words “especially on the first day”. Perhaps however it is only coming to include the second day in the Diaspora and not Chol Hamoed.
 See Hiskashrus
 See Piskeiy Teshuvos 658/11; Minchas Yitzchak 5/65; Kaf Hachaim 658/66
 Ksav Sofer 129 brought in Biur Halacha 658 “Lo Yitnenu”; Minchas Yitzchak 5/65
 Piskeiy Teshuvos ibid