According to them the sequel- perhaps to be called The Children- would have been set "in the present"- that is to say in the 1880s- a couple of decades on from the events of the Brothers Karamazov. Ivan and Dimitri were no longer to be central characters and Alyosha- now a political radical- was going to attempt a terrorist outrage- perhaps the assassination of the tsar- and then suffer execution. Also projected was his career as the village schoolmaster, marriage to Lise and an affair- this really is a bolt from the blue- with Grushenka.
Much of this is a little shocking. But reading the existing novel in the light of the intended denouement one picks up all sorts of little hints and foreshadowings. Grushenka's fascination with Alyosha is going to have long-term consequences. Lise's character arc will be completed. The startling change of tone that comes with the sudden prominence accorded Kolya Krassotkin makes sense if he is to figure in the sequel as Alyosha's political disciple and right-hand man.
Do I regret that The Children was never written? Of course, of course. Would it have complemented the existing novel or diminished it? Who can say.