I don’t think people are concentrating into a small number of countries at all. I don’t know where you get your numbers but according to IOM international migration was 280.6 million in 2020, of which 33.8 million were refugees. Most refugees stayed in the geographic region (e.g. a tenth of the entire refugee destination was to Turkey followed by Jordan at 3 million).
True normal migration lead by culture, love, opportunities, etc. is a lot bigger factor in migration then the climate crisis, but those are not the people suffering from hostile border policies (well not on the same scale as refugees) and don’t need the same help to bypass said policies. But it would be foolish to dismiss climate change and not expect the proportion of refugees to go up in the next years compared to other migrants.
Contrary to popular belief, migration is not caused by climate, it's caused by culture.