If the pandemic is a testament to how countries will respond to climate-related crises, the assumption that countries will turn increasingly inward seems reasonable. 2020 marked the lowest point in resettlement efforts in the past two decades, making the concept of global solidarity something largely feigned under the guise of scientific collaboration. Fortunately, Biden has increased the American quota for resettlement to 125,000 (from the historic-low of 15,000 imposed under Trump).
There was also a ruling by the UN last year that governments cannot return people to countries where their lives may be in danger from the effects of climate change - a huge step considering that the concept of "climate refugees" is still quite recent and not well-defined. In practice, that could look like what you've described, with thousands of people arriving at the border of another nation in search of refuge.