There is a recurring narrative (especially but not exclusively in left-leaning circles) that today's "developed nations", particularly Western former colonial powers, owe much of their wealth and progress to the exploitation of their former colonies. This is frequently used to counter the idea that the "First World's" headway is due to any inherent superiority, either in culture or social organization (such as capitalism, market liberalism or democracy). The implication, apart from the call for reparations that often follows, is that today's "developing nations" are unlikely to ever catch up, replacing the idea of global progress with one of eternal injustice (barring a revolution, which of course is what many proponents of this narrative are aiming for). However, to my untrained eye, the current development of many "developing countries" seems to falsify this idea. Is that so? I assume some work has been done on this -- can anyone here point me in the right direction? That would be greatly appreciated.
To complement @rocinante answer the argument described in the question silently assumes that growth can happen only if you steal from somebody. So if the past colonies cannot steal from anybody, they are bound not to ameliorate their material welfare.
Assume two persons, $A$ and $B$, that currently have the same unexploited resources. It so happens that in period $t$ (which may last for decades or centuries- "colonialism"), $A$ manages to ruthlessly exploit $B$, and appropriates a large portion of $B$ resources, which, together with $A$'s resources (and the industrial revolution), create a growth drive for $A$.
Now move to period $t+1$, "post-colonialism". $A$ is currently rich due to past exploitation of $B$, and $B$ is poor, perhaps in a worse situation than he was in the beginning of period $t$. Still, this is a different state of affairs than the one holding at the beginnings of period $t$, when both $A$ and $B$ where poor with unexploited resources, and the industrial revolution had not happened.
$B$ may now benefit from the current status of $A$, thus creating a "growth drive" for himself. For example, $B$ can become the main outsourcer for $A$'s business.
This is up to a point what is happening, and so this specific fact cannot be used in support of any ethical approach to the matter. The situation may still be described by some as "eternal injustice", but "eternal injustice" is a totally different thing than "eternal economic inequality".
For example,assume that $B$ eventually catches up with $A$. Still, one could argue that because $A$ made his riches by stealing from $B$, while $B$ made his riches by working for $A$, the situation is still unjust, even though economically they currently are at the same level of material well-being.
The recent development of many "developing countries" and "emerging markets" does not mean that Western colonial powers did not exploit them in the past.
Emerging economies owe their modern GDP growth to cheap labor, as (former) Western colonial powers switched their (expensive) manufacturing bases to countries like China, India, etc. In the past, they were exploited mainly for their natural resources moreso than labor.
The continued GDP growth of Western colonial powers is due to a shift from manufacturing (labor-intensive, low-education professions) to "high tech" industries which require significantly more education (medical research, computer programming, etc.).
The book "Capital" by Thomas Piketty has a far more detailed history of how capital flows shifted from the 18th century to modern times, and the economic interplay between Western colonial powers and the developing nations. https://en.wikipedia.org/wiki/Capital_in_the_Twenty-First_Century
The growth of developing countries has been accompanied by rising resource prices and crimping of growth in the First World. This does not mean that anyone is stealing resources per se. It shows that resources are scarce and constrain growth, and one countries' demand makes it harder for other economies to buy oil.
The sort of commodity price behavior that we see during rapid growth in the Third World does provide support for a mercantilist, zero-sum, resource constrained view of growth. This does have elements of colonialist/resource imperialist model where the First World invades oil producers to privatize the resource rents.