In a much larger Talmudic discussion (Eruvin 40b) that involves the question of whether one must eat/drink after saying a blessing (instead of having someone else, such as a child drink), as well as the propriety of giving alcohol to children, Tosafos record two stories, which are instructive as to how people conduct themselves with reference to eating and drinking on a fast day.
I share because when there is a bris on a fast day (10th of Tevet, referenced below, is coming up), this information is instructive.
תוספות מסכת עירובין דף מ עמוד ב
ומעשה היה בחופה בעשרה בטבת נתנו הכוס לתינוק לשתות [ומעשה היה בברית מילה ברבינו יעקב בר יקר שחל עשרה באב בא' בשבת והוא היה אב"ד וצוה להתפלל מנחה גדולה ורחצו ואכלו מפני שי"ט שלהן היה כדכתיב שש אנכי כו' והאי דלא מברכינן שהחיינו משום צערא דינוקא תוס' שאנ"ץ].
When we say a blessing over wine on a fast day, we have the custom to give the wine to the child - which at a bris is the baby. This may or may not be the source for our giving wine to the baby at every bris anyway."It happened that there was a wedding on the 10th of Tevet (a fast day), and they gave the cup (of wine) to a child to drink (the word "Tinok" means baby - but sometimes refers to a child).""It happened that there was a bris with Rabbi Yaakov bar Yakar which fell on the 10th of Av which fell on Sunday (meaning it was the day they were observing the 9th of Av fast, pushed to Sunday). He was the Head of the Rabbinic Court. He instructed everyone to pray the early Mincha, and then to wash and eat, for it was a Yom Tov for them, as it says, 'I rejoice at the fulfillment of Your word...'And the reason we do not say "Shehechiyanu" at a bris is because of the pain the child experiences."
But the points we take from these tales for bris practices on fast days are the following:
1. The easiest person to give to drink on a fast day is the baby
2. A fast day which is pushed off, even a fast day as important as 9 Av, is a day when participants at the bris may eat. While this is not a common practice, it is mostly due to lack of awareness of the rules.
3. A baby does feel pain at a bris. This is not something we celebrate through the recitation of Shehechiyanu. [This is a confirmation that this mitzvah is one of the rare mitzvos in the Torah that has pain associated with it (the other being fasting on Yom Kippur). This is why numbing the baby is advocated, while some argue that the speed in which we operate serves as its own numbing.]