Women are exempt from the mitzva to take the lulav since it is time-bound, and as a rule, women are exempt from time-bound positive mitzvot (Kiddushin 29a). Nevertheless, a woman who wishes to fulfill a time-bound positive mitzva is rewarded for doing so.
According to most Sephardim, women do not recite a berakha over the performance of a time-bound positive mitzva, for how can one who is not commanded recite a berakha that contains the word “ve-tzivanu” (“and commanded us”)? According to Ashkenazic custom, since women are fulfilling a mitzva, they recite the berakha; the language of the berakha is not a problem, as they do not recite “and commanded me” but “and commanded us” – “us” connotes the Jewish people as a whole. Even though Sephardic women generally do not recite berakhot over time-bound positive mitzvot, many recite the berakha over the lulav, and some have offered kabbalistic reasons for this.
Once a young boy knows how to properly shake the lulav, forth and back, up and down, his father must train him to do so. When the son has reached an age when he can go to the synagogue and pray, his father should buy him his own set of four species, so he can shake the lulav at the times ordained by the Sages. If a father does not have the means to buy his son his own set, he should allow his son to use his, so that the son can fulfill the mitzva (Sukka 42a; SA 657:1; MB ad loc. 4).
Young girls should be encouraged to shake the lulav each day of Sukkot. Even though women and girls are exempt, they fulfill a mitzva by doing so, and it teaches them to love mitzvot.
. As we learned above in 4:13, on the first Yom Tov one cannot fulfill the obligation with a borrowed lulav. Thus, one must be careful not to give the lulav to a minor then. Since he is unable to give it back, no one else would be able to fulfill the mitzva with it. However, after all the adults have fulfilled the mitzva, one may give the lulav to a minor (SA 658:6). Some say that even then, it is preferable not to give the lulav to a minor, because an adult might come along who needs to fulfill the mitzva (Eliya Rabba ad loc. 10). According to this, presumably the minor would use a borrowed lulav. However, some say that a minor does not fulfill the mitzva for training purposes on the first Yom Tov unless the lulav belongs to him, just as adults cannot fulfill the mitzva with a borrowed lulav (MA; Eliya Rabba; Pri Megadim; Ḥayei Adam). Others say that since this is for training, he may use a borrowed lulav and may even recite the berakha beforehand (Bigdei Yesha; Mordekhai, Raavan, and SA according to MB 658:28 and SHT ad loc. 36). This opinion may be relied upon.