The first mistake when approaching immigration is to take all the different types of immigration and even non-immigration and lower the level of detail to simply "immigration". This means:
1. refugees
2. legal immigrants
3. non-immigrants magic beam of
3.1. temporary workers ──infinite──> "iMMiGraNTs"
3.2. tourists irrationality and "mIGrAntS"
4. students
5. illegal immigrants
6. stateless
7. prisoners of war
8. etc
If you want to fix a problem you need to divide it into subproblems and solve each problem, not mix everything together and make a larger problem harder to solve.
Mixing everything together is obfuscating the problem which is the modus operandi of those who want to prevent the solution of the problem as they have an incentive to keep the problem around.
Politicians like keeping immigration a problem because it gives them a way to divide people into groups and rewarding those groups depending on their political needs, making you easier to rule. Which is what they do for everything else. In their case, you are the problem to solve, not the actual problem they so passionately talk about. And when you get angry you play right into their villain claws.
The brain gain/drain dynamics between countries is real. Brain gain is what you definitely want as a country.