Funnily enough, the current "immigration" fiasco in the UK is about the Windrush generation, who came over from Jamaica at the instigation of the UK which was running the country at the time. The UK is trying to retroactively undo this and deport them, including people who were born in the UK, despite the fact that they originally came over on British passports.
Running a country for someone doesn't mean you need to give citizenship, residency, or even visas to them. It just means you do the math and see that it's obviously less suffering to remove some liberty and self-governance to improve the lives of millions. Over time, slowly give back bits of governance while retaining the right to override. Then take it from there.
It takes some mental gymnastics to allow a situation where innocent people live in fear and can be robbed and killed without reprisals only to claim you've delivered "human rights" and democracy.
The trouble with this is that, as currently demonstrated around the world, you need to keep killing increasingly large numbers of people to maintain this level of control. The argument for apartheit South Africa was always this, and it was a terrible idea. It was the argument for tanks on the streets in Northern Ireland and the argument for tanks on the streets of Hungary in 56.
It doesn't matter because it doesn't justify apartheid, just shows that you don't need to murder lots of people. Having bad countries outsource key government roles to good countries doesn't need to involve any sort of tiered citizenship or segregation.
Current murder rates seem high and are likely underreported. (For instance, Guatemala won't record a homicide if the person doesn't die right away but is helped by aid personnel.)
Further, apartheid isn't a case of what I'm proposing. They were running their own country for their benefit, not as a humanitarian/client setup.
For many countries sending people to prime immigration target countries, the only visible fix is running their countries for them.
Instead, the UN and other groups do the opposite. They run around singing about human rights, enabling lawlessness and crime to go on unchecked while decent citizens become frustrated, some of them escaping and contributing to immigration.
In particular, Central America. I know many people that would be 100% onboard with defending and cleaning up their country. Instead, they all tell me they don't even carry a weapon, because using it will get them jailed with a short life expectancy.
Meanwhile, pretty much every Guatemalan I speak with pines for the days of Rios Montt which kept things safe and clean. Yet is decried by the international community which doesn't have to live with the result of their choices.
Fix the source, take a heavy hand. There's no need for the West to babysit people.
Letting a tiny amount of people in to your own countries will accomplish little to actually relieve suffering. It's pointless and costly and not a real solution.