We are seeing it in Africa now and the northern migration in the Americas is also at least partly due to climate. The sheer numbers involved will dwarf previous migrations and overwhelm some countries.
Over population and dysfunctional governments leads to migration. Climate change is not a significant factor in migration pressures from Africa or South America since these pressures would be happening even if climate change was not an issue.
Trying to blame developed countries for cultural and political problems in developing countries is simply a means to extort money.