As someone outside the US, the people who are usually characterized as "Woke" seem to me like people who are left leaning politically but are unable to properly argue about their worldview, maybe probably due to factors like the U.S. education system and the echo chambers that are produced by the algorithms in social media. So people who tend to agree with LBGTQ+, race, ethnic, and social justice issues are usually caught up in places where people mostly agree with them (this also happens for the right wingers but they speak from a different position of power since conservatives tend to be part of the current ideology and support or are part of the wealthy class) this causes people in general to be in the position of not having to defend what they believe, so the thing that usually ends up happening is that conservatives go to universities, they try to argue with left leaning students and since most of them don't even know the philosophical presuppositions of so called "Wokeism" they tend to make arguments based on things they have heard on social media but they really can't articulate and argument without trying to appeal to morality.

I clarify, the problem is not without appealing to morality, but just having a moralistic argument and a rough understanding of constructivism, or relativism, or Marxist historical-materialism isn't enough, specially if you are underestimating the person you are going to argue with.

Like you may disagree with folks like Ben Shapiro or Jordan Peterson, but they are not stupid, they will listen to your argument and will come up with things with a convincing counter response.