I think this kind of case goes far beyond typical family dynamics into a whole different realm. I have commented repeatedly on this forum, that IMO narcissism, although demonized on the internet, is a pretty ordinary human trait, technically it's a personality disorder', whereby someone tries to manipulate other people who are alive and who stay alive.
Personality disorders - Symptoms and causes
Whereas, when someone moves into actually killing people, their narcissism or paranoia or borderline PD or whatever, becomes as irrelevant as which party they vote for. IMO, killing people off is something far beyond what we encounter in life, that's why it's so shocking and fascinating.
I think understanding this case will involve looking at other similar cases where several people have conspired to murder off other people. Frankly, I can't think of any that are similar. IMO, it doesn't resemble cases where someone hires a pair of hit men to eliminate someone else. It doesn't resemble cases where people get into extreme sexual sadism with their victims, or an 'us against them' attack on established institutions and authority. There doesn't appear to have been the hatred that was behind the Pike County massacre of one family by another.
Folie a deux are familiar, where a couple are so close they share a mental delusion that leads them to do things that one person would not do on their own. So, perhaps this is a 'Folie a trois', that the murders would not have happened except through the strange chemistry and shared delusion of these three people coming together at a particular time in their lives.
I don't think anyone was under the mind control of someone else, I think each of them had motivations and personalities that meshed into some kind of shared delusion that doing these horrible things was a good idea, or at least, necessary. For example, I don't think Alex was a helpless puppet of his sister or of her lover. IMO, assuming he was the one who did the actual killing of the children, he must have had some killer instinct that was fulfilled in committing these actions, because he didn't just do it once, and then be racked with trauma and regret, he did it twice.