But with mass immigration, it becomes everyone's problem.
For the country with very high fertility rates and overpopulation, eventually it will become so impoverished people will die from starvation and famine, being left in droves on the wayside.
The breeding becomes unsustainable, and eventually the government comes under pressure to create a clampdown on reproduction. This has already happened at one in point in India and China. Other countries have resorted to incentives to get people sterilized, like Indonesia.
If the people from country A have the right to immigrate into country B, then country B has the right to demand (or enforce if necessary) population control in country A.
Otherwise, even if the majority of countries get their population growth under control, it won't matter, because masses of population will still continue pouring out of the handful of impoverished countries that don't have a handle on their growth.