Answer:
Some countries that have a very high GDP per capita, e.g. Qatar which has the highest of all, are generally considered developing nations and their Human Development Index is really bad. In the case of Qatar, United Arab Emirates, Saudi Arabia, and the rest of middle eastern oil exporting countries, the wealth of the country is only held by a very small percentage of the population, usually less than 10%. That 10% is extremely rich and the other 90% are extremely poor and live under very harsh conditions.
The European Union's experience is a clear example of how an increase in the flow of workers affects economies. Generally speaking, an increase in the flow of workers will shift to the right the long run aggregate supply (LRAS) curve and the production possibilities frontier (PPF). This right shift means that the total production output will increase due to a decrease in production costs.
In the short run, if the flow of migrants was too large, the economy may experience some negative effects due to the increase in the aggregate demand curve and the increase in government spending. But this type of events are not common, not even the Syrian refugees caused such a disruption. The last time such an event happened was during World War II, and that was a major unprecedented event and nothing similar has happened again.
Any increase in spending is offset by a decrease in production costs and an increase in total production output. Something similar happens to the 25 million Mexicans that live in the US, a lot of them illegally, but they do not hurt the economy, they help it grow.