So looks like this will be a story/anime about three...kids...Ok the three are girls.
The girl in the back with the bow in her hair is Inuyasha and Kagome daughter Moroha.
Purple ponytail is Setsuna she is Sesshomaru's daughter has an older sister...I'm guessing she also the daughter of that girl that trailed after him; Rin. Who works for Kohaku Sango's brother.
Short hair girl in a suit is from the present. Towa Higurashi... who is also Setsuna's sister...so she is also Sesshomaru's daughter. She was raised by Kagome's little brother.