To be perfectly truthful children have lived in tiny homes since the dawn of man.
During the Ice age humans lived in either naturally formed caves or tents made of mammoth skins. It was no surprise to find generations living in the same small space. By 4,000 BC farming had spread across Europe and when families committed to the lifestyle they more or less stopped living in tents made from animal skins and began to live in huts made from stone or wattle and daub with thatched roof the huts were typically one room with a sort of fire pit in the center and sleeping pads around the perimeter. Just after the Roman Empire The Saxons lived in wooden huts with thatched roofs. Usually there was only one room shared by everybody. (Poor people shared their huts with animals divided from them by a screen. During the winter the animals body heat helped keep the hut warm). Presumably though the question is more about the modern tiny house movement.
While a modern tiny house in the style of Tumbleweed Tiny’s or some other tiny house on wheels is typically less then 250 square feet it is seemingly ample room for a small family. For further reference take a look at the Berzin’s family of 4 documented on Tiny House Family, the Odom’s family of 3 as seen on Tiny r(E)volution, the Helgerson family of 4 in Portland, or even the Jordan family of 3 who collectively write 320 square feet home. While there are challenges they are unsurmountable. It takes discipline, the proper attitude, preparation, and the desire to live closely and honestly. It also helps that many tiny houses are located outside of heavily urban areas which gives a great resource in the outdoors to which to escape for space and freedom.
Of course the success of multiple people living in a tiny house lies in its design. Typically speaking a single level 200 square foot sheep’s wagon is not conducive to a family with children. However, a 240 square foot house with two sleeping lofts, a bathroom with a door, etc. is quite well suited for a family. Again, coupled with the outdoors and even the attendance of school and/or work and the recipe can be one for great success.