My theory is that wokescolds are actually essentially people with a "conservative mindset" in a liberal disguise.

(note: I'm not as well-read on Haidt's newer liberty/oppression moral foundation so I'm not including it here)

Haidt says liberals use "harm/care" and "fairness" as their foundations, while conservatives add "authority", "ingroup", and "purity". It's notable that people in the "woke" subset appeal to those foundations as well:

Purity is somewhat obvious; words and actions have taken on a strong association with being "pure" and "impure". Policing speech used to be associated primarily with conservativism (think of religious people getting upset at swearing). Now it's also associated with an ever-changing list of 'acceptable' and 'unacceptable' terms (I still remember the short time when everyone was saying "trans" and they would get upset if you didn't use the "")

Ingroup is reasonably obvious too; there is an extremely strong "us vs. them" mentality among the woke. There are also signaling games—think of how many woke people use explicit visual signals to demonstrate they're one of the "good guys".

Appeal to authority is a little harder; things like the hero-worship of RBG or Stacey Abrams resemble it, but woke culture is just as willing to rip its idols to shreds in the way traditional conservatives may not.

So my theory is that because of the tribalism that's developed in the US political sphere, there are people who should be conservatives, but can't for social reasons (all of their friends and community are ostensibly liberal), so they channel their conservative moral foundations into other avenues.

There are other ways this crops up too, relevant to other research in political psychology. The traditional liberal is characterized by a general optimism about humanity, while the traditional conservative is skeptical and more defensive. Think of a stereotypical liberal hippie perhaps naïvely seeing the best in everyone compared to the stereotypical conservative suburbanite, cautious or afraid of people unlike him.

The woke, though, are observably pessimistic towards other people. The mantra of everybody being racist is cut from the same cloth as the stereotypical conservative idea that everyone is out to get them.

I'm actually really curious what the endgame is here. I kind of expect a realignment at some point, where the cadre of people who hold actual liberal moral foundational values, even if they're at odds with the woke, eventually becomes more of a cultural block. Most of the people that started on the (cultural) left (like myself) have been very reluctant to publicly reveal their adherence to these values due to the social risk, but it feels like we're at a tipping point, and I've had more and more conversations where I'll be talking to someone one-on-one and they'll eventually reveal that they're a little skeptical of some of the woke dogma.

Regarding where the woke end up, I don't see them aligning with traditional conservatives any time soon—even though they have similar moral foundations, there's just way too much object level stuff they disagree with to see eye-to-eye. But I think this is actually par for the course for conservative mindsets. While all people with liberal mindsets naturally glom together (tribalism is alien to their values), "purity", "authority", and especially "ingroup" encourage multiple factions within those with conservative mindsets. For an extreme example, think of the Yugoslav wars, which were essentially fought between multiple different conservative factions, not conservative vs. liberal.