What are Reinert's reproaches to the Ricardo theory? Economists accuse me in vulgarization of their science, so I'll edit the text from the very beginning to remove the inaccuracies.  
Main question
I have just read the book by a norwegian economist, Erik Reinert, "How Rich Countries Got Rich… and Why Poor Countries Stay Poor". I think it is very interesting for people who want to understand how modern economics works, and, in particular, where poverty comes from. And, what is unexpected, the problem, as Reinert it presents, seems to be not so complicated from the point of view of a mathematician.  
Reinert blames in everything the Ricardo theory of comparative advantages.  He writes that it is exactly this theory that justifies poverty in the modern world. He gives numerous examples, when arguments based on the Ricardo theory made some countries extremely poor, without hope for correcting the situation in future. He paints horrible pictures of the situation in Mongolia, Peru, Equador, Haiti, Russia, etc. Simultaneously, from what he writes it becomes clear why there are so many migrants in the modern world, so many jobless people, so many refugees, etc. -- so I recommend his book to everyone.
My main question is what I wrote in the title:

What are Reinert's reproaches to the Ricardo theory?

Ricardo examples
This question was in the previous version of the text, and since nobody answered yet, I think that this must be difficut. That is why I want to add another question that I believe is much easier and can clarify the situation: 

In which extent the Ricardo examples (without further theory, just examples) explain the problems?

This needs explanation. First, I foresee new accusations in vulgarization from economists. So I think it will be useful to remind that vulgarization is a standard trick in mathematics: when people are trying to understand a problem, they often remove something from the theory (or replace this detail with something else), and look what happens then. A classic example is the Parallel postulate: in attempts to understand what is wrong Lobachevsky replaced this axiom in the theory and looked what happens after that. In modern mathematics this trick is widely used, since all axiomatic theories are based on this idea. 
Second, I have to explain to mathematicians what is meant by the Ricardo examples. 
The classic Ricardo example (I change the numbers for simplicity). Suppose we consider two countries, $A$ and $B$, and each of them produces two goods, $G_1$ and $G_2$, and the table of expenses is as follows:
$$
\begin{matrix}
 & G_1 & G_2 \\ A: & 1 & 1 \\
B: & 2 & 4 \\
\end{matrix}
$$
(this means that in the country $A$ one unit of $G_1$ costs 1 man-hour, and the same for $G_2$, and in the country $B$ one unit of $G_1$ costs 2 man-hours, while one unit of $G_2$ costs 4 man-hours). 
Both goods, $G_1$ and $G_2$, are cheaper in the coutry $A$, so it seems evident, that it is  more profitable to produce both $G_1$ and $G_2$ in the country $A$. But Ricardo notices that the difference in the comparative expences changes the situation: if the country $A$ conveys  the production of $G_1$ to the country $B$, while $B$ conveys $G_2$ to $A$, and they begin to trade, this becomes more profitable for both countries, since


*

*in $A$ each man-hour still gives one unit of $G_2$, but when selling it to $B$, $A$ takes two units of $G_1$ (instead of one unit of $G_1$, as it was when they did not trade),

*in $B$ each man-hour still gives $1/2$ unit of $G_1$, but when selling it to $A$, $B$ takes $1/2$ unit of $G_2$ (instead of $1/4$ of $G_2$, as it was when they did not trade).
This justifies the following observation

If in a given country $A$ the expences in the production of a good $G_2$ in comparison with another good, $G_1$, is less than in a country $B$, then it is more profitable (for both countries) to produce $G_2$ in the country $A$ (and to trade).  

The examples like this do not exhaust the Ricardo theory, of course, but I suspect that the essential effects can be explained with their help. As illustrations I suggest to consider the following two situations. 
Example #1. Consider three countries with two goods and with the following table of expences: 
$$
\begin{matrix}
 & G_1 & G_2 \\A: & 1 & 1 \\
B: & 2 & 4 \\
C: & 4 & 2 \\
\end{matrix}
$$
Here, according to Ricardo, 


*

*the least comparative expenses for producing the good $G_1$ are in the country $B$, so the good $G_1$ must be produced in the country $B$,

*at the same time the least comparative expenses for producing the good $G_2$ are in the country $C$, so the good $G_2$ must be produced in the country $C$.
And the Ricardo trick gives the conclusion that 

the country $A$ must abandon the production of both $G_1$ and $G_2$. 

Example #2. In the Ricardo example nothing prevents us to consider workforce as another, third commodity in this situation, and to look at the comparative expences in its production. We can just change the unit of measure, and use the good $G_1$ instead of "man-hours", then the table of expenses becomes the following:
$$
\begin{matrix}
 & \text{workforce} & G_2 \\A: & 1 & 1 \\
B: & 1/2 & 2 \\
\end{matrix}
$$
(this means that in the country $A$ one man-hour costs one unit of $G_1$ and the same for one unit of $G_2$, and in the country $B$ one man-hour costs $1/2$ unit of $G_1$, while one unit of $G_2$ costs 2 units of $G_1$). 
And the Ricardo trick gives the conclusion that

the country $A$ must abandon the production of its workforce (i.e. make all its citizens jobless, and import the workforce from $B$).

Qualitative differences
This trick with the Ricardo examples seems to be an easiest way to explain the problem to an outlooker. That's why I ask specialists what I wrote above: in which extent these examples explain the situation? My question is, if it is possible that the general theory prescribes qualitatively the same conclusions as in these examples (i.e. the calculations just change the quantitative answer from $0$ to a little $\varepsilon>0$, and that's all)?  In particular,


*

*Is it possible that in Example #1 the countries $B$ and $C$ take off all (or most of) the production from the country $A$, even if in $A$  it is more effective?

*Is it possible that in Example #2 the country $A$ comes to a situation when it must discharge all (or most of) its citizens, even if the production there is more effective than in the other countries?
Reinert's criticism inspires this suspicion. 
Experiments
One more question: 

Do economists use experiments in this field? 

(I asked this in economic forum without success.) 
A: This answer is essentially an elaboration of Will Sawin's answer. It also comes with the caveat that it comes from an economist. It is easier to give an answer by generalizing the problem. There are $l$ commodities, including inputs (labor is treated as a commodity for this purpose). A production plan is an $l$-tuple of real numbers with negative entries representing inouts used and positive entries outputs being produced. If a production plan is feasible, it means that using the negative part of the $l$-tuple as inputs, one can produce the positve part of the $l$-tuple as outputs. Now assume that country $C$ has the nonempty production set $P_C\subseteq\mathbb{R}^l$,listing all feasible production plans for the country. If countries work together, they have an aggregate production set $P=\sum_C P_C$. $P$ represents what all countries are jointly able to produce from some initial inputs. The sign convention allows the ouput produced by some country being used as an input somewhere else. To complete the model, we add for every country $C$ an initial endowment $e_C\in\mathbb{R}^l$. We let $e=\sum_C e_C$. Then $e+P$ is the net output all countries can produce. 
Now if all commodities are desirable and coordination between the countries works well. The countries will jointly produce a point in $e+P$ that is maximal under the usual vector ordering (there are known conditions for the compactness of $P$, so existence is not an issue). Let's call such a production plan technologically inefficient. It was Ricardo's insight that such a production plan may involve production by countries that are "absolutely disadvantaged". Many economic models of trade explain trade patterns by the resulting specialization. That Ricardo's theory is still used means nothing more than that.
Now a model of functioning communist planning will lead to the respective coordination to produce a technologically efficient plan. But this is also the outcome of an idealized model of a market economy, the Arrow-Debreu-McKenzie model. This model extends what I have just sketched and is taught in graduate microeconomics courses around the world and can be found in basically all standard textbooks. It includes consumers who come with personal endowmentss and preferences and can decide on their consumption plans. This increases the explanatory power and reduces the degrees of freedom (the resulting equation system may still not have a unique solution). 
There is also a small group of economists, the neo-Ricardians, who prefer models without consumer demand, because the additional degrees of freedom allow them to give explanatory power to class struggle or other social phenomena.   
A: There is an extensive literature on Ricardian models with many goods and countries. For a survey which also answers some of the questions you make in your later edit, see:
Eaton, Jonathan, and Samuel Kortum. 2012. "Putting Ricardo to Work." Journal of Economic Perspectives, 26(2): 65-90.
https://www.aeaweb.org/articles.php?doi=10.1257/jep.26.2.65 (freely accessible full-text)
In a Ricardian world there is no unemployment by assumption. What ensures full employment is that wages need not be equal in all countries. The level of wages adjusts in response to productivity differences until a point is reached where each country is the cheapest supplier of some commodities to some destinations (which might include itself). Which commodities these are for each country is determined by the Ricardian principle of comparative advantage.
Original research publications:


*

*Two countries, continuum of goods: Dornbusch, Rudiger, Stanley Fischer, and Paul Anthony Samuelson. "Comparative advantage, trade, and payments in a Ricardian model with a continuum of goods." The American Economic Review (1977): 823-839.  http://www.jstor.org/stable/1828066 

*Many countries, many goods: Eaton, Jonathan, and Samuel Kortum. "Technology, geography, and trade." Econometrica (2002): 1741-1779. http://www.jstor.org/stable/3082019 
Ricardian theory has many simplifications: only one input in production, constant returns to scale, perfect competition, full employment, no dynamic effects on technology and resources … Weakening these assumptions may reverse the conclusions that Ricardo reaches and economists have come up with many scenarios where free trade may not be beneficial. 
But the papers cited above show that under Ricardo's assumptions it is possible to give a fully rigorous mathematical account of the theory with no logical problems while preserving many of the qualitative features of the 2-country, 2-goods case.
A: You misinterpret the Ricardian theory even in the case of two countries and two goods. In your first example, $G_2$ should be produced in $A$ and $G_1$ should be produced in $B$. But this doesn't completely explain production - it doesn't say whether also some $G_1$ should be produced in $A$ or some $G_2$ should be produced in $B$.
The only possibility that Ricacardo's theory eliminates is that both some $G_1$ is produced in $A$ and some $G_2$ is produced in $B$. Then there are gains from trade. Every other type of production is not ruled out.
The problem is that you are missing some of the information. Production is determined by supply and demand, and your model fails to include the demand. If we know how much of each good is needed, and how much is needed at each cost, then we can compute optimal production. This will happen even in your cases with more goods and more countries. I guess the only exception will be cases of complete indifference, say if two countries have exactly the same costs to produce each good. (Note that the demand here is not necessarily a capitalist concept. It could represent a market economy where the demands of those with more money is counted more, or a socialist world government accounting for the desires of each of its citizens equally, or a single international project like building the LHC where fixed amounts of given goods are needed to complete a single goal.)
Ricardo's orginal model did not need to consider demand because he only needed to show that there were some gains from trade, not determine precisely who should produce what.
So: 
Think of production as continuous, or at least discrete with many possibilities - each country does not produce just one good.
Use information about demand to find the right amount of production of each good in each country.
A: For answers to your mathematical questions about how to extend the Ricardian theory to multiple countries and/or multiple goods, start with Will Ethier's paper on higher-dimensional issues  in trade theory.
For a better understanding of the role of models in economic theory (with specific regard to the Ricardian trade model), see Krugman or Suranovic.
