Step 1: There’s a serious problem that a lot of people have. Mostly, people ignore it, or assume that there’s nothing they can do about it.
Step 2: People start complaining about the problem, expressing their frustration, etc.
Step 3: Groups start organizing around the problem. People propose moderate, reasonable solutions and start lobbying for them. Mailing lists are created, signs are made, pamphlets are distributed, and people start identifying as Pro-Xers.
Step 4: A big debate starts, Pro-Xers vs. Anti-Xers. Both groups duke it out, trying to win supporters and fight for their cause.
Step 5: The Pro-Xers win their first major victory, and everybody cheers. A few moderate Pro-Xers see that X is winning, and decide to spend less time on X, becoming passive supporters. Meanwhile, other people see that X is winning, and decide to become Pro-Xers to get status and power.
Step 6: The problem starts to get better (or at least, people think it’s getting better). This makes the Pro-Xers popular and wins them lots of support.
Step 7: The Pro-Xers win more victories. At each step, the leadership becomes more radical, and more of the moderates retire and become passive supporters. The Pro-X leadership starts becoming corrupt, using their high status to do favors for their friends, and so on.
Step 8: Either out of partisanship or ignorance, the media doesn’t notice the steady radicalization of X, and assumes that the current Pro-X positions are similar to the old, moderate Pro-X positions. All of the old moderates continue to support X due to halo effect.
Step 9: X is now a toxic group. The leaders of the Pro-X movement are a mix of radical extremists and self-interested power brokers, and X is so influential that they’re now effectively dictators, with people bending over backwards to give them what they want.
The Rationalist Conspiracy 
That sounds plausible enough, but it’s not clear to me how often it actually happens, either in absolute terms or as a fraction of instances of toxic politics. Is it based on actual analysis of lots of real cases?
Definitely happens out in the wild. I find that description disturbingly similar to what happened to the German Pirate Party. Though it was not extremists of the Pirate flavour (anarchists, maybe?) that took over, but SJWs, mixed with the inevitable attention whores.
The first thing I thought of when I read this was the evolution of the ANC in South Africa: while I think the description in this post fits quite well, it’s worth noting that the corrupt leadership often leads to offshoot young turks (in this case Julius Malema and the EFF).