Children fasting on Yom Kippur:
Below age 9-10-No fasting: Children who are less than nine years old [for a healthy child, and less than ten years old for a weak child] are not to be oppressed on Yom Kippur, being this can lead to danger. [They are to be fed as normal, and it is even forbidden to delay their meals from their regular times.] Even if the child desires to be stringent upon himself [and fast], he is to be protested [and forced to eat] in order so he does not enter into danger. Nevertheless, a child may only be fed for his own sake and not for the sake of fulfilling a Mitzvah. [The above is from the letter of the law, however many are accustomed to permit children to fast for a certain amount of hours, even if they are below the age of nine. It is however forbidden to force them to fast even for a few hours, even at night. Thus, if they ask to eat or drink they are to be allowed to do so.]
9-10 Years old-Delay Meals: Beginning from the age of nine for a healthy child, and ten for a weak child, both boys and girls are to be educated to fast on Yom Kippur, through having their meals delayed from their set time. One is to delay the meal an hour or more from its set time based on the amount of time delay the child can handle. Thus, if the meal is normally eaten at 3:00 pm they are to eat it at 4:00 pm or later. [Likewise, at night the child should fast. However, if the child is very thirsty, he may be given to drink even at night.]
11 Years old-Fast: It is disputed whether a child above the age of 11 is Rabbinically obligated to fast on Yom Kippur. Practically, although the main Halachic ruling follows the stringent opinion, nevertheless, one may be lenient in a case that the child is weak and is not strong enough to handle the fast, even if there is no danger involved if he were to complete the fast. Based on this, people today are no longer accustomed to educate their child to fast the entire day of Yom Kippur even after 11 years old, until they become Bar or Bas Mitzvah, as today all children are considered weak. Nevertheless, if one knows for certain that the child is healthy and strong enough to sustain the fast, then this leniency does not apply. [It is proper that the child fast until at least midday.]
From Bar and Bas Mitzvah: When a child has reached the age of 12 years old for a girl and 13 years old for a boy, then if he or she has grown two pubic hairs, they are considered adults and are obligated in all the commands, including the fasts. If the children have reached this age, but do not have two pubic hairs, they must nevertheless guard all the commands out of doubt that perhaps the hairs already grew and later fell off.
Obligation of the father to reprimand his child for not fasting: The obligation of educating a child in the above is on the father. Thus, if the father sees his child disobeying the above, he must reprimand him.
Yaleh Veyavo in Birchas Hamazon: A child who eats a Kezayis of bread on Yom Kippur, is to recite Birchas Hamazon. In Birchas Hamazon, one is to add Yaleh Veyavo prior to Uvinei Yerushalayim and say “Beyom Hakippurim Hazeh”. [However, some Poskim rule that based on Admur in the Siddur one is not to recite Yaleh Veyavo in Birchas Hamazon. Practically it is to be recited.] When Yom Kippur falls on Shabbos, he is to also add Ritzei.
 See Admur 616:5-11; Piskeiy Teshuvos 616:2; Nitei Gavriel 36
 Admur 616:10; Michaber 616:2; Rambam Yom Kippur 2:11
 Meaning, one does not have to delay their meals. [so is implied from Admur ibid as otherwise a child before nine and over nine would have the same law. Hence, one must learn that a child before nine is not oppressed at all even to delay a meal.]
 As is implied from the ruling that if the child desires to fast [or delay the meals] he is to be protested. Thus, certainly one may not force the child to fast. [ibid] Vetzaruch Iyun why a 2:3 hour delay would lead to danger?
 Admur ibid; Rama 616:2; Kol Bo 69
 See also the laws of Tisha B’av chapter 556 Giving a child to drink from the wine is also not done, being that children may only be given to eat on Tishe Beav for their own sake, in order not to accustom them to eating on Tishe Beav even when they get older. [M”A 556] Thus we see that even a child who may be fed, may only be fed for his own sake.
 Ashel Avraham Butchach 616; Chikrei Lev 109; See Piskeiy Teshuvos 616:2
 Alef Hamagen 616:2
 Admur 616:5; Michaber and Rama 6161:2; Mishneh Yuma 82a
 See Kitzur SHU”A 133:19
 Yalkut Yosef brought in Piskeiy Teshuvos 616:2
 Admur 616:6 and 11
 Admur 616:6; Michaber and Rama 616:2
The dispute: Some Poskim rule that when the child reaches 11 years old, he must begin to fast for the entire day. This obligation is a Rabbinical decree, and applies to both males and females [1st opinion in Admur 616:6; Michaber 616:2; Rambam Yom Kippur 2:10; Rif; Rav Nachman Yuma 82a] Other Poskim however rule that until the child reaches Bar/Bas Mitzvah, there is no even Rabbinical obligation for him/her to fast the entire day, and they only are obligated to fast by delaying their meals. [2nd opinion in Admur ibid; Rama 616:2; Rebbe Yochanon Yuma ibid; Bahag; Maharitz Geios; Tosafus Kesubos 50a; Or Zarua 2:278; Rokeiach 217; Yireim 51; Hagahos Maimanis 2:10]
 Admur 616:6;l Rama ibid; Terumas Hadeshen 155
 Admur 616:11; M”A 616:2; Bach 616; See Machaatzis Hashekel ibid; Elya Raba 616
The reason: 1) Today the children all learn Torah, and the learning of Torah weakens them [We find this concept also regarding the Onah obligation that a Torah Scholar need only have relations once a week, being that his strength is lessened due to his Torah learning] and 2) Today people are weaker than previous times being that “weakness has descended upon the world, and thus a regular child above 11 is not considered strong enough to fast. [Admur ibid]
 So is implied from end of Admur 616:11; and so rules Mishneh Berurah and Piskeiy Teshuvos 616:2
 Yalkut Yosef brought in Piskeiy Teshuvos 616:2
 Admur 616:8; Michaber 616:2
 Admur 616:4
 This applies to all laws which a child is required to be educated in, as explained in chapter 343.
 Admur 618:18; first opinion in 188:11
Other opinions: Taz 618:10
 Shaar Hakolel 34:9
 Piskeiy Hasiddur 185; See however Hiskashrus who writes it is to be recited in Harachaman, See Nitei Gavriel Bein Hametzarim 66:11