I posted a few pieces (here and here) recently on genetics and growth. The Economist even picked up on Justin Cook's work on lactose tolerance and development. Justin's work on both lactose and the HLA system are about very specific genes, while the other research I mentioned is about genetic heritability of certain behaviors associated with growth, without specifying any particular genes.
There is another line of research on evolution and growth pioneered by Oded Galor and Omer Moav. They propose that natural selection over different types of individuals could have led to the onset of sustained economic growth. In particular, they focus on selection over preferences for the quantity and quality of kids. This is very much the second kind of research I mentioned above; it does not identify some specific gene that matters for growth, it suggests a mechanism through which selection could have operated. The original paper is linked here, but they have a nice summary article here that explains the logic without all the math.
Let's be careful about terminology here. Evolution in general requires both mutation and natural selection. GM is really about natural selection, not mutation. They take as given the presence of two types of people in the population. "Rabbits" like to have large families, but do not invest much in their kid's human capital. "Elephants" have a few kids, but invest a lot in those kids. Their theory is about the proportions of those types change over time due to economic forces, and eventually how a rising prevalence of Elephants leads to a speed-up in technological change. Yes, at some point there must have been a mutation that led to the differentiation between the types, but we can think of that as happening well back in history. They don't propose that some mutation occurred at some specific year or a specific place to make this all work.
How does the underlying logic work? In the early Malthusian period, with very low income per capita, the Elephants actually have the evolutionary advantage. Why? In the Malthusian world, everyone is so poor that higher income leads to higher fertility no matter your type. Each Elephant kid has high human capital, and thus relatively high fertility compared to Rabbits. So the proportion of Elephants tends to increase in the population. And a higher proportion of Elephants means that average human capital is rising over time.
As the human capital rises, so does the pace of technological progress. At first this doesn't do much, as the growth of technology is not sufficient to overcome the force of Malthusian population pressure. But eventually there is high enough human capital that technological change happens so rapidly that people reach the upper limit on fertility rates, and choose to spend any additional income on increasing their kids human capital rather than having more kids. This is the tipping point where human capital and technological change go into a virtuous cycle. Higher human capital leads to higher technological change, which leads to higher human capital, etc.. etc.. and you have sustained growth. Once this occurs, the relationship of income and fertility flips to become negative - the richer you are the fewer kids you have, just the opposite of the Malthusian period. This flip in sign is not unique to their explanation based on natural selection, the same type of flip is central to the general unified growth model in Galor and Weil.
After this transition point, the evolutionary advantage also flips to Rabbits. Why? Because the fertility rates decline with income, and as Elephants are richer due to their human capital, they have fewer kids than Rabbits. So Rabbits begin taking up a larger and larger proportion of the population. But everyone is already relatively rich, so this doesn't mean that human capital levels are low generally. There is sufficient human capital to sustain technological progress.
Do we know if this exact mechanism is what generated sustained growth? No. To establish that you'd have to identify the precise genes that govern preferences for quantity/quality of kids and show that they varied within the population over time in a manner consistent with the GM model. But there are little bits and pieces of circumstantial evidence that work for GM. Greg Clark's Farewell to Alms documents his research showing that in fact richer families tended to have more kids in pre-Industrial Revolution England. This fits with the selection mechanism proposed by GM. Similarly, Galor and Marc Klemp have a working paper out on the reproductive success of families in 17th and 18th century Quebec (a place and time with particularly detailed records), and the data shows that it was families with moderate fertility rates that actually had the most kids in subsequent generations, not those with the higher fertility rates. Again, it fits the selection mechanism proposed by GM for the Malthusian era.
Note that even if it isn't true genetic differences in preferences for quantity/quality, you still need to have selection working for population composition to matter for sustained growth. Let's say that quantity/quality preferences are purely cultural, passed on from parents to kids imperfectly but with some fidelity over time. Then the GM mechanism could still hold up, but it would be the cultural spread of preferences for high quality that generated the take-off, not the spread of specific genes.
There are reasons to be skeptical about this explanation, just as you should be skeptical about any hypothesis. But don't dismiss it on the basis that natural selection moves far too slowly for this to have mattered for human populations. Galor and Moav have a number of very telling examples regarding the speed of selection within populations over just a few generations. The classic story is peppered moths during the Industrial Revolution. Peppered moths tend to be white, with little black spots on them - hence the name. But there are black varieties. With the rise of coal in the UK black moths became far more prevalent, as they were harder to spot for predators against the blackened sides of buildings. Within a few years the population jumped from predominantly white to predominantly black. And then flipped back to white when clean air regulations came into force. Given the variation in the population already exists, natural selection can take place very quickly to change population composition. So imagining that human population composition could change substantially over hundreds or thousands of years is reasonable.
Last, does GM mean that generating growth in poor countries is doomed to failure because their genetic composition is "wrong"? No. GM is a story about the rise of sustained growth at the global level. Suggesting that poor countries need to get their genetic mix right in order to grow is like suggesting that they need to adopt steam engines and telegraphs before they can step up to gas engines and mobile phones. The question of how to catch up to the frontier is an entirely different question than explaining how we got a frontier in the first place.