Will everyone in the world eventually become the same, mixed race?
Yes, most likely. There will be some holdouts in the isolated tribes of the Amazon or the multitude of Andamanese islands though.
It depends on what "eventually" and "mixed" means. If "eventually" means in a thousand years then evolution will do its thing and people will more and more intermingle, but it will take a long time. If "mixed" means 50-50 then this will also take a long time, if ever. But if "mixed" means, say <1%, of another race, then this has happened already for most people in the US already!