Opening the borders is absolutely the quick patch to the issue, but the long-term compounding of the issue to the point of catastrophic failure. While "wealth" may not be a zero-sum concept, planetary resources are absolutely zero-sum. It is possible to use these resources more efficiently, but there is a limit.
If a region's population has outstripped it's resources, it is up to that population to reduce it's own population. I do not advocate killing people, but I do advocate population controls.
Limiting the number of children to 2 per person can almost guarantee the population will reduce itself to a sustainable level.
1. No new people are allowed in (closing borders)
2. Once a man or woman has had 2 children, they are medically prevented from having any more.
3. No fertility treatments are permitted (prevents attempts at having triplets and more at once)
*If triplets or more are expected, there should be no penalty. I do not advocate forcing someone to terminate a fetus because it would put them over the limit.
While I believe in the above statements, in my country it would never come to pass because a person's individual freedom outstrips long-term planning. (there is also a large number of people that would argue that it violates their religious freedom)