The following is only a quick sketch and is not meant to capture everything. I hope though that it helps some readers get a handle on the steps by which the liberal worldview unfolds.
In brief, the modern liberal mindset gets to where it is through these steps:
Step 1: A denial that there is a stable, objective, external truth that exists or can be known to us.
Step 2: Therefore, the good in life is the act of choosing for ourselves; this is what generates the value that gives human life its dignity. The core value is a freedom understood to mean a right to determine our own self and being (self-determination and individual autonomy).
(A variant on this: the good is not in a community disciplining itself to a truth that resides eternally outside of itself but in the assertion of its own will or in its collective ability to shape its own reality.)
Step 3: If human dignity rests on a freedom to determine our own self, then the unchosen aspects of self should be made not to matter, including our biological sex, race and sexuality and the social roles flowing from these.
So, although for liberals there is no moral order existing objectively outside the individual, there is a new morality in which we must accept the right of others to self-determine. Therefore, we must be non-discriminatory, open to the other, tolerant, inclusive, accepting of diversity and so on.
Step 4: Therefore, there does exist a kind of moral order generated by liberalism, which does then restrict what the individual may or may not do or believe (despite the starting point of a belief in a freedom to self-determine). Individuals in a liberal order may not have beliefs that would make the unchosen aspects of self (biological sex, race, sexuality) or the social roles flowing from these matter in a public context. An individual may be judged guilty, in thought or deed, of discrimination, intolerance, exclusion, prejudice, sexism and so on.
Step 5: If there is no objective, external layer of reality by which we might come to know moral truths or ideals, then the informal, traditional ordering of societies will appear "opaque," irrational or baseless. Instead, it will seem more efficient and scientific to order society along technical lines, according to what is required by markets (i.e. society is organised around the tangible and measurable criteria of what helps an economy to be profitable) or along bureaucratic lines, in which the individual stands in a direct line to the state, with the goals being again the measurable and tangible ones of health, education, housing, welfare and so on.
Step 6: In practice modern societies are ordered along both market and bureaucratic lines. However, some moderns (right-liberals) prefer the market option (the economy) and others (left-liberals) the bureaucracy (the social option). The concept of liberty for right-liberals will emphasise a deregulated freedom to act in the market; left-liberals will prefer a more interventionist state guaranteeing an equality in those conditions of life which help individuals to act autonomously (i.e. they want to overcome disadvantage, understood to mean access to resources for a self-determining life, particularly disadvantage that correlates to those unchosen qualities that are not meant to matter such as our biological sex or race).
Step 7: Happiness and equality don't arrive as they're supposed to. Why? Some liberal moderns assert that it is because some groups of people have a self-interest in preventing it happening. Some groups of people get together and construct a false identity (e.g. maleness or whiteness) which then allows them an unearned privilege at the expense of other groups who are positioned as "the other" (the non-privileged groups). The whole of society is set up to uphold this unearned privilege, meaning that the privileged groups and the society they inhabit must be deconstructed before the promised society of equality and freedom can finally arrive.
So those are the basic steps. There are plenty of people who are at least dimly aware that there is something wrong with the liberal modernity that results from this way of thinking about the world. For instance, liberal societies aren't able to maintain borders - they are porous. In liberal societies, men and white people are attacked as oppressors. In liberal societies, politics is reduced to the management of the economy and certain social goods such as education and housing. In liberal societies there is no encouragement toward positive character goals or virtues, there are only the passive goals of tolerating or not discriminating. In liberal societies, important aspects of identity, such as those relating to manhood and womanhood, or to ancestry and ethny, are gradually dissolved.
But people feel overwhelmed by what appears to be a monolith. How can you possibly take on something that seems to have entrenched itself so deeply into Western society?
Those who do try to resist usually choose the wrong step. Last century most social conservatives, for instance, chose step 6. They opposed the left-liberal wing of modernity by supporting the right-liberal one. Predictably, it had little effect.
There have also been alternative forms of modernity which took the more collectivist path as described briefly in step 2 (possibly under the influence of German idealist philosophy). But this did not challenge modernity itself: it was a different face of modernity rather than a rejection of it.
It is possible as well to challenge liberal modernity at step 7 by objecting to the vilification of white people and men. This is a useful thing to do (a matter of self-defence), but it won't stop the juggernaut of modernity as it doesn't challenge it at its source.
Finally, there have also been those who have challenged modernity by asking what led to step 1. Was it the nominalist victory over philosophical realism? The influence of Gnosticism amongst influential intellectuals? This is an interesting and useful discussion but I don't think it's the best entry point for resisting liberal modernity.
My opinion is that an effective challenge will include two things. First, stating as clearly as possible what liberalism is and where it leads (as a means of clearing people's minds of it as thoroughly as possible). Second, reasserting what was lost from step 1 (so that we shunt aside centuries of liberal development in one decisive move).
The second step will require some scaffolding, as it is now difficult for people to readily give voice to pre-liberal thought. We can start a discussion about character and virtue; about ideals of manliness or womanliness; of natural forms of identity and attachment, and of the loves, loyalties and duties associated with them; of what makes for a good life, or of what some of our life purposes might be; of how the human personality comes to fruition or to its higher purposes; of how different sides of the human personality might be harmonised or ordered.