I don’t believe this is true. Statistical projections like that are based on simplistic formulas, eg “there were 10m Muslims in 1990, 20m in 2000, 40m in 2010, so in 2020 there should be 80m and in 2060 1280m”. By that logic, there will be over 40bil in 100 years, just in Europe, which is a lot more than the total population the planet can feed.
Even though it is true that Turks (for example) currently breed faster than Germans, and that there are more and more Turks in Germany (and there will be a huge influx once Turkey joins the EU), there is a certain limit to how many will even want to go there, and many will eventually go back home. Greeks have been moving back to “the homeland” from America and Australia, and the East Germans who went West are already thinking of going back just 20 years’ later. The same will happen to all the Poles in UK and Ireland after a while. Of course there will be large numbers of immigrants left in the host countries, and this mass population migration is nothing new (how do you think the Celts ended up in Scotland? Or the Mongols in Hungary, and the Persians in Finland?).
If you go back enough, everybody originally came from somewhere else. And all of them started off from Africa.
So basically I think the change will be smaller than people imagine, and smooth enough for us to deal with it. Yes, our world will change. It changes constantly. But that’s always been the case, and people whose ancestors left Kenya 200,000 years ago should stop attacking people whose ancestors left Kenya 199,000 years ago.