Pop in Q

The plot of Pop in Q is strongly similar to the 90's anime because it involves school-girls from different high schools being transferred from their world to another world to save it from disaster. Pop in Q involves dancing while Magic Knight Rayearth uses magic and weapons to help everyone. Moreso, it even emphasizes character development to help further the plot. 

The themes in those two anime are common in 'A schoolgirl travels to another world by accident' although the differences between those two are that Fushigi Yugi's another world is based on Ancient China and Pop In Q's another world has cute animal like fairy species living in small houses. Fushigi Yugi and Pop In Q focused on character development where the main protagonist has to help everyone in the another world while confronting her own problems as well. What they do to save the world: In Fushigi Yuugi, the protagonist has to summon Suzaku, the god of the south and Pop In Q: The protagonist teams up with another few protagonists to save the another world by dancing. read more 

PriPara and Popin Q both involve girls who can transform into idols. Some episodes of PriPara, as well as one of the movies, involve having to save the world of PriPara with music.