It is a frequent claim but I think this asks the wrong question. From Are Liberals Dying Out? by Hrishikesh Joshi and Jonny Anomaly. The claim is that Liberals in the modern American sense of Democrats are characterized by low rates of family formation and even when they do form, have low rates of fertility. Therefore, the argument proceeds, liberals will eventually be outproduced by conservatives.
Like all bad ideas, there is a core element that is true. Democrats indeed do have lower family formation rates and lower fertility rates.
Beyond that, there are all sorts of problems. The biggest is definitional. What is a liberal? I consider myself a classical liberal with libertarian tendencies: I take much inspiration from Locke, Smith, Hume, Hayek. That scarcely qualifies me as a liberal in terms of today's nomenclature. You can see this evolution of terminology in other countries such as Australia where the conservative party is the Liberal Party who contest elections against the Labour Party.
Today's British Conservative Party, in terms of policies, looks a lot like the American Democratic Party circa 1980.
So names don't really tell us much. The author's acknowledge the problem of definitions but try to skirt around it. They make a feint at trying to rescue their position by alluding to the psychological trait of Authoritarianism but they fail to tie that to either end of the spectrum in terms of evidence. Indeed, in the US right now, the groups with the most overt manifestation of authoritarianism by far are clustered at the liberal end of the spectrum - BLM, Antifa, Postmodern critical theorists on university campuses and in the media, etc.
So definitions is the biggest rock on which this founders. But it is not a single rock reef. Where are Democrats concentrated? In cities. Might that be the reason that Democrats have a somewhat lower fertility rate? Occam's Razor would suggest so.
I suspect the real question of interest is not whether Liberals are being outbred.
I think there are two behavioral attributes that are negative and it would be interesting to know whether we are seeing a decline in those behaviors.
My first question would be "Is self-destructive fanaticism dying out?" It is fanatics, not liberals, who are garnering all the negative headlines. BLM rioters, Antifa brawlers, postmodernist obstructionists, hecklers and authoritarians, ISIS, Taliban, Al Qaeda, etc.
It is easy to get lost in the headlines and the availability bias. Things do seem to be falling apart, the center does not appear as if it can hold. But The Better Angels of Our Nature by Steven Pinker makes a solid case that in fact violence and fanaticism are declining. Sure, it pops up here and there. More to the point, elevated fanaticism is more prevalent in some regions than others. But overall it is declining.
My second question is more challenging. "Is existentially destructive ennui increasing?" A question to which I do not have a good answer.
You can have a falling population because of increasing violence but you can also have a falling population because of enervation and ennui. That is what we are seeing in western Europe and Japan. Populations no longer seem to have as great a commitment to the great chain of being, to a future beyond their own life span.
This could be due to an all pervasive consumerism where children take away your opportunity to consume. It could be owing to a loss of religious faith with its commitment to an ongoing future through children and a sense of intergenerational commitment. It could be due to some combination of any number of things.
I suspect that self-destructive fanaticism is indeed self-destructing and Darwin Award-like, removing itself from the future gene pool and therefore reducing the behavioral propensity towards fanaticism in the future.
Ennui and enervation is modern western classically liberal societies may actually be the more serious threat to existential destruction. I don't think it is inevitable but I think that is the issue to focus on.