How do we explain the significance of fasting even on minor fast days to our children of chinuch age?
Obviously they are not required to fast, but how do they internalize the seriousness of the day? Our Sages have the answers.
This topic covers Chinuch laws pertaining to fasting on MINOR fast days.
1. There are 4 minor public fast days throughout the year – 3rd Tishrei (Fast of Gedaliah), 10th Teves (Asaras B’Teves), 17 Tammuz and 9 Av (Tisha B’Av).  and  both address only 17 Tammuz and 9 Av.
Although no mention is made about 10th Teves and 3rd Tishrei in both books,  addresses it as follows on 10th Teves:
“It is proper to train children to give them only the amount and type of food that is necessary and thereby diminish the pleasure that they have so that they will experience the sense of mourning of the community.” addresses it as follows for Laws Concerning Public Fast Days in Chapter 121:9:
“Children, too, although they do not have to fast, nevertheless, if they understand the significance of mourning, it is proper to train them, and give them only bread and water*, so that they will be part of the communal mourning.
* They may be given any other simple food (Ibid. 550:5)
Therefore, I consulted  and  and extracted the laws of fasting for minors, and believe they apply as well to 10th Teves and 3rd Tishrei.
2. Children below the age of bar/bas mitzvah are not required to fast on any fast-day, with the exception of Yom Kippur.  
3. There is one restriction which applies to children who have reached the age of understanding on the concept of mourning (i.e. 7 or 8 yo depending on the intelligence of the child) on all fast days. Parents
should only feed simple foods to these children on these fast days. Delicacies or specially tasty things should be avoided. In this way, the child is educated to take part in the fasting of the adult community. No such restriction applies to children too young to understand what a fast day is. 
4. Some have a custom, nevertheless, to train children who have reached the age of chinuch to fast part of the day if possible.