No. Classical liberals oppose the part of the status quo dubbed 'wokeism.' They think it's illiberal & regressive & want neoliberalism
Left liberals share this perception of wokeists, but don't like neoliberalism much & want progressive social policies AND freedom. (Cont) https://twitter.com/Jose1Pino/status/1322643705493270528

"Right-wingers" are a mixed bunch coz they include those who prioritise free markets & capitalism and those who prioritise tradition, family & faith. They tend to be critical of capitalism & consumerism which they see as empty hedonism. Neither like wokeism (cont)

Then there are liberal conservatives who are often referred to as 'centre-right' (some overlap with classical liberals here too) and they generally support progress but prefer to go more slowly & cautiously than left-liberals.

People who want "regression to hierarchical systems" are best understood as 'reactionaries' than 'conservatives. Then there are also right-wing populists who are their own weird postmodern thing that lives in a world of simplistic feel-good narratives. (cont).

José correctly says "Leftists' want progress. However, Marxists will usually say that leftism refers solely to them & progress is economic equality. They don't usually like the wokeists either or accept them as leftists seeing them as bourgeois elitists dividing the working class

Marxists tend to blame left-liberals for wokeists (even though they are illiberal) & some liberals & many conservatives blame Marxists for wokeists (even tho they don't advocate for the working class which is partly why the WC increasingly votes right now).

And then we have what is known as 'wokeism' or 'social justice' & this is an identity-politics based leftwing ideology that draws on postmodern ideas of knowledge, power & language more than either Marxist or liberal views but can include elements of both to various degrees.

You can also find wokeists who call themselves Marxists and speak against capitalism in the same way Marxists do but they usually combine this with typical wokeism & say things like 'Anti-racism is also anti-capitalism' or 'feminism is incompatible with capitalism.'

There are also feminists who actually are Marxists and not at all woke or have Marxian ideas and oppose wokeism usually in the form of trans activism & they are definitely left but often accused of being indistinguishable from right-wing transphobes by the wokeists.

So, no, the situation is not that we have right-wingers who want society to regress & leftists who want society to progress. Social reality is much more complicated. We can't always factor in all the distinctions I just covered but ppl with very simplistic takes are best ignored.