Once children reach the age of ḥinukh – the age when they can understand the mitzvot of Yom Kippur – we teach them not to wash, apply ointment, or wear shoes on Yom Kippur. Generally, children reach this stage at the age of five or six. Some go beyond this and make sure their children do not wear shoes from the age of three.
In addition to the mitzva to train children to keep the mitzvot of the day, it is also forbidden for adults to cause children (even day-old babies) to transgress. Just as adults may not feed children insects or blood, or cause a young kohen to become impure, so too, it is forbidden for adults to wash children, apply ointment to them, or put shoes on them (Peninei Halakha: Shabbat 24:2). However, if there is a certain degree of medical need, one may wash a child and apply ointment. These do not fall under the prohibition of taking medicine (which is a rabbinic prohibition on Shabbat and Yom Tov), because the Sages permitted undertaking these activities for a child who is sick or experiencing discomfort (ibid. 24:6). Similarly, if a child is likely to hurt himself going barefoot, an adult may put on his shoes for him.
In terms of fasting, we do not train children who are only five or six to do so, because they are not strong enough, and fasting may be harmful to them. Therefore, we wait until the age of nine. At that point, healthy children are encouraged to fast part of the day. For example, if they generally eat breakfast at eight in the morning, they should wait until the afternoon to eat. Weaker children should begin fasting at age ten rather than nine.
From the age of eleven, children should be encouraged to fast the whole day. If they are weak, they can be lenient and fast only half the day.
Girls from the age of twelve have a Torah obligation to fast, while boys have a rabbinic one. Even a twelve-year-old boy who is weak should make an effort to fast the whole day. If he is sick (even if not deathly ill), he is not obligated to fast, since he is not yet thirteen. Nevertheless, he should try to fast until the afternoon. From the age of thirteen, boys, too, have a Torah obligation to fast.
Many encourage younger children who have reached the age of ḥinukh to fast through the night. Even though some object to this stringency, many follow it in order to train the children to participate a little in the fast. However, if the children ask to eat or drink, they should be fed (Elef Ha-magen 616:5).
Many maintain that before the age of nine, children should not be allowed to fast at all during the day, lest they endanger themselves (Rema 616:2). However, most children want to fast for a few hours even before they turn nine. Since doctors do not feel that this is dangerous, most people let them fast during the morning. We are not required to try to stop this custom (Eshel Avraham [Buczacz], based on Rashi).
According to Rif, Rambam, and SA 616:2, there is no difference in the age at which boys and girls begin training for the fast. Healthy children start fasting part of the day at the age of nine, and weaker ones start at ten. By the age of eleven, all healthy children are encouraged to fast a full day. Only children who are sickly should wait to fast until their Torah obligation sets in. (This second position follows a different understanding of R. Naḥman.)
Others follow R. Yoḥanan, who maintains that we never train minors to fast a whole day. Only when their Torah obligation sets in do they fast a full day. Training to fast part of the day begins two years before halakhic adulthood. This is the position of R. Yitzḥak ibn Gi’at, Roke’aḥ, and Yere’im, as well as AHS 616:17 and Halikhot Shlomo 6:14. Eliya Rabba suggests that the reason that le-khatḥila, children should not fast for a full day until they reach majority, is that all children are considered sick. In contrast, Terumat Ha-deshen and Rema state that we rely upon R. Yoḥanan only if a minor is weak and not strong enough to fast.
MB 616:9 cites the various positions. R. Eliyahu in Ma’amar Mordekhai Le-mo’adim U-leyamim 45:49 writes that minors should be encouraged to follow the ruling of SA. This is what I write above. The exception is an eleven-year-old who is weak, where I follow the lenient position, as this is the common practice.