It is forbidden to instruct a child to perform a Melacha (forbidden action) on Shabbat, regardless of whether the action is needed for the adult or for the child. This applies even if the action is forbidden only Mi’de’rabbanan (by Rabbinic enactment, as opposed to Torah law).
Although it is forbidden to feed a child non-kosher food, it is permissible to feed a child kosher food at a time when eating is forbidden for adults – such as before praying Shaharit in the morning, and before Kiddush on Shabbat.
During the Nine Days before Tisha B’Ab, when it is customary to refrain from meat, it is permissible to feed children meat. Since eating meat is halachically permissible during the Nine Days, and we refrain from meat only by force of an accepted custom, meat may be fed to children during this period, even to children who have already reached the age of Hinuch (halachic training). This is the ruling of Hacham Bension Abba Shaul (1924-1998), in his work Or Le’siyon (1:38), where he adds that the reason we refrain from meat during this period is to mourn our inability to offer sacrifices after the destruction of the Bet Ha’mikdash, and this reason is not applicable to minors, who in any event cannot bring sacrifices.
Hacham Ovadia Yosef (Hazon Ovadia – Ta’aniyot, p. 245) rules that a child who has not yet reached the age of Hinuch may be bathed in hot water as usual during the week of Tisha B’Ab, even though adults refrain from bathing in hot water during this period. Hacham Bension Abba Shaul rules more leniently, allowing even older children to bathe in hot water during the week of Tisha B’Ab, until they reach the age of Bar or Bat Misva.
If Tisha B’Ab is observed on Sunday, a child who has reached the age of Hinuch should recite Habdala before eating on Tisha B’Av.
Minors are allowed to take haircuts during Hol Ha’mo’ed, even though this is forbidden for adults.
Summary: One may not have a child perform on Shabbat an action forbidden for adults on Shabbat. During the period before Tisha B’Ab, when we abstain from eating meat and bathing normally in hot water, children may be fed meat and may be bathed in hot water. When Tisha B’Ab falls on Sunday, a child who has reached the age of halachic training should recite Habdala before eating on Tisha B’Ab. Children may be given haircuts on Hol Ha’mo’ed.