Rav Menahem Azarya Mi’Pano (Italy, 1548-1620), in one of his responsa (111), wrote that there is no value whatsoever in forcing children under the age of Misva obligation to fast on the fast days. To the contrary, one should ensure that children are properly fed on fast days. This is also the view of Hacham Ovadia Yosef, who ruled that even children aged 11 and 12 must eat on fast days, even if they want to fast. Hacham Ovadia noted that particularly regarding Tisha B’Ab (and the other fasts commemorating the events surrounding the Hurban), there is no purpose in training children to fast, as we fervently hope that the Mashiah will come and there will no longer be any obligation to fast. There is certainly no need to train our children in the Misva of fasting if we hope that they will never have to observe this Misva.
If a child insists on fasting, his parents may allow him to delay or skip his breakfast, but he should not be allowed to miss lunch. This is the ruling of Hacham Ovadia Yosef and of Rav Shelomo Zalman Auerbach (Israel, 1910-1995).
An interesting question arises in a year when Tisha B’Ab falls on Shabbat, and is thus observed on Sunday (as it is this year, 5772), and a boy turns 13 on the 10th of Ab. Is he required to fast, or is he exempt, since he was a minor on the actual day of Tisha B’Ab? Hacham Ovadia Yosef ruled that if the boy feels weak and would have difficulty fasting, he may be lenient and eat on that day (listen to audio recording for precise citation).
Another interesting situation arises in the case of a woman who gave birth one month before Tisha B’Ab. Within thirty days of birth, a woman is exempt from fasting, and in fact is not allowed to fast. However, if a woman gave birth exactly thirty days before Tisha B’Ab, such that the thirty-first day is the 10th of Ab which falls in Sunday and is thus observed as Tisha B’Ab, she must fast. Even though the day before she would have been exempt from the fast, she must fast on Sunday since she is no longer within thirty days of childbirth.
Summary: Children under the age of Bar Misva should not be allowed to fast on Tisha B’Ab; if they wish, they can skip breakfast, but they should not fast beyond lunch. A boy who becomes Bar Misva on Sunday, the 10th of Ab, which is observed as Tisha B’Ab, does not have to fast if he feels weak and will have difficulty fasting. If a woman gave birth thirty-one days before Sunday, the 10th of Ab, she must observe the fast.