Beis Yaakov has been a Chasidish/Litvish institution and system right from the founding until today. They aren’t mixed because there’s so many girls k”h that they need many Beis Yaakov schools. As such, many Chasidishe kehilos each have their own Beis Yaakov school. But, for example, Beis Yaakov of Boro Park, which is the oldest and largest Beis Yaakov today, is a very thorough mix of Litvish girls and Chasidish girls from many different chasiduses.
Nowadays even some Sefardic girls schools call themselves Beis Yaakov.