Globally, Pew Research notes that "babies born to Muslims will begin to outnumber Christian births by 2035":
This top headline gets a lot of press. What is less looked at are the drivers of rising Muslim birth rates.
In an average year, Christians and Muslims both have about the same number of babies worldwide: ~210 to 220 million. Christians are often the majority in "older" population regions, however, and their average age is older: there are a lot more deaths among Christians than Muslims. In an average year, Christians have about ~100 million deaths, vs Islam's ~60 million. This means the net demographic increase for Muslims is much higher. Any upward tick in the Muslim birth rate will, therefore, have sizable effects.
Both religions are being impacted by regional trends:
Unfortunately the two reports on which the data is based, "The future of World Religions: Population Growth Projections, 2010-2050" and "The Changing Global Religious Landscape" are somewhat limited reports. While they explore demographic patterns in detail, they (1) in my estimation seriously underestimate % Christian in some places (such as China, estimated at 5% Christian), and (2) seriously underestimate the role of "religious switching" (conversion). The reports state they modeled "religious switching" in only 70 countries; they do not "model" (count) religious switching in either India or China. This is a significant flaw: the population of Christians in both places has grown primarily through switching over the past several decades. Further, the report goes on to estimate the growth with and without switching, and concludes switching makes no appreciable difference (but I doubt this outcome because of the flawed methodology of only modeling switching in 70 countries). (I also find it humorous that Pew's "Future of World Religions" report, on p. 187, says "Since religious change previously has never been projected on this scale..."--completely ignoring the work of the World Christian Encyclopedia/World Christian Database, which projected religious populations for every country and every religion to 2050 nearly two decades ago).
So, while this "headline" on Muslim vs Christian births gets a lot of press, I'm not giving it a lot of credibility. The numbers between birth rates are so close and the factors driving them--and their calculation--are so variable and uncertain, that I think the headline has just as good a chance of being untrue as true.