The Ottoman Empire was extremely tolerant of other religions by the standard of... well, basically whatever century you're talking about (except the 20th century -- the Ottoman tradition failed spectacularly in the 20th century). When Spain expelled the Jews, for instance, the Ottomans offered them refuge (and sent ships). This sort of tradition continues for centuries. While the Ottomans never really did consider Muslims and non-Muslims to be perfectly equal (as evidenced by the fact that, whenever a case came to the Sikayet the faiths were always defined), it made precious few attempts to force people to adopt their faith. The only real exception I can think of is the devsirme, where Christian children were collected as a "tax," converted to Islam, and given an education. The end result of that life was either the imperial administration or the Janissaries (their rivals, the Sipahis, were thoroughly Turkish).

As for the demographics, that came in three parts. The first was the Ottomans, had a fairly Muslim-dominated elite. And it was often made clear that, to the Ottomans, Muslims were generally perceived as being equal regardless of ethnic origin. The Koprulu family (Albanian Muslims) secured myriad high-level civil posts that would've been nigh impossible to achieve if their members weren't Muslim -- but the fact that they were ethnic Armenians doesn't seem to have been a terrible burden. Conversion to Islam opened doors and that's about as far as the Ottoman Empire went in trying to actively convert its subjects.

The second factor would be the Muslim migrations. Those would just be the large migrations. That's not counting the slow trickle of freshly converted Muslims that were the result of the devsirme system (fun fact: Khayr-ad-Din, the most famous naval commander in Ottoman history, was the son of a Janissary and, together with his brother, conquered half of North Africa and then turned it over to the sultan). And, of course, it's not counting the inevitable population shifts that went hand-in-hand with Ottoman conquests (peasants would often flee, most notably in the case of Constantinople, prior to the Ottoman conquest; Ottoman soldiers, especially irregulars like the akinci, sometimes took the opportunity to settle the now-vacant land).

Finally, and perhaps most overtly, is the series of population transfers that took place after WWI. Here's a pretty simplistic visualization of what that looked like. That was only a part of the population transfers to "clean up" ethnic borders in the aftermath of WWI (not pictured are similar movements of ethnic Bulgarians, for instance). And, tragically, the Armenian Genocide also served to indirectly convert parts of Anatolia to Islam.

Those are the broad strokes. Hope that helps.