Natsu no Hi to, Bokura to, Mahou

Four children - Mikami, Sakurai, Souma and Hiiro - are bored during their summer holidays. Mikami kicks his shoe down a flight of steps, where it lands in front of a girl. She picks it up as he tries to apologize, but instead of returning it, the girl runs off with the shoe.
The children chase after her, but before long they find themselves in an unfamiliar town. Their mobile phones are out of range, and the old man they try to ask about their location has never seen a pair of trainers before, and everyone's clothes, hairstyles and actions, as well as the layout of the town around them all look old-fashioned.
The reason? They've traveled back in time, to 1959...
But why have they been brought to this period in time?