Why should it?
That’s asinine, you’re treating periphery countries like they’re glorified breeding-stock for the developed world’s work-force.
Edit: To make my point more clear, the whole reason why developing nations have higher birthrates than developed ones is because they’re developing/underdeveloped. They lack access to contraceptives, and substantive access to women’s healthcare; and they also oftentimes have economies that still rely to some extent, or a large extent on non-mechanized smallholder, or subsistence agriculture. That, or they otherwise have social institutions that allow for, or require children to enter the workforce. This means that having children in those countries is often an economic boon to a family (because they can contribute to household incomes through work), and avoiding having them can be very difficult for women.
If you solve their problem of being underdeveloped, & hyper-exploited (which you should be doing if you’re a “queermunist”), then that means that they are likely also going to be in a position where they have declining birthrates because there will no longer be an object material incentive to have children, and women who don’t want to would be able to prevent it.
The idea of shoring up a declining population “through immigration” only works so long as you have an underdeveloped periphery of peoples who want to come flock to the West, or to developed nations in search of higher wages & a higher standard of living (or just avoiding Imperialist political meddling), rather than staying at home.
Well, no, birth rates are low because the reproduction of labor is unpaid labor. Yes, development is associated with lower birthrates, but only because no developed country has ever seriously tried to make reproductive labor a real job. Doing so would decrease the size of the workforce for production of commodities.
Now you’re totally right that the people migrating from the Global South are fleeing underdevelopment from imperialism, and that this is itself a factor of underdevelopment. What you haven’t considered is why the imperial core limits migration.
Racism is part of it, but only part of the larger structural base. If they allowed unlimited migration the imperial core would be filled with people from the periphery as they flee underdevelopment. This would at once reduce the availability of labor in the periphery and raise the contradictions of imperialism by making peripheral concerns into domestic concerns.
Migrants influence the society they’re part of, causing agitation against imperialism. This would, ultimately, destabilize the core and allow for development to resume without imperial meddling.