The reason that birth rates are down is because of freeing of women. Women have found out, that they do not need a man to be successful and their whole reason for existence is not to be the man's birthing machine.
You can see a direct link between women's suffrage, access to education and healthcare and falling birth rates in basically any society. Of course economic issues have a barring also, but that is more a consequence of women's suffrage, access to education for women and so on.
Take Spain since the evidence here is striking. During the right wing fascist dictatorship of Franco, the law of the day was very conservative and very pro Catholic Church. There was no access to abortion or contraception, women were discouraged from attending higher education and even basic schooling and the expectation of society for women was to have babies and as many as possible. Hell for a time way up into the 1970s it was "bad" for women to have pants on. Healthcare for women was poor and as was diets. This in turn meant in turn the classic Spanish woman being small in height and often wide. Why? because boys were more desirable and were seen as the "money" earners even at a young age, and hence were given better and more food. This in turn meant girls got less and their natural growth was hampered. On top of that, girls started giving birth at a young age which dont help in the growth area (well expect for the boobies of course).
Now fast forward to the 1980s and democracy in Spain. Since then, the number of girls attending school has gone up and the number of girls attending higher education has sky-rocketed. Like in other more "modern" European countries, girls are now a majority of certain traditionally male dominated fields at university.. doctors and lawyers for example. Because of this, girls are not getting married at an early age like under Franco and do not see their sole role in life to be married and make babies. This means women in Spain wait later and later to get babies, which in turn means they only get one or 2 instead of 5 to 9 as under Franco. This means the birth rate goes down.
At the same time overall health of the nation has increased as the wealth has increased and guess what... the height of the typical Spanish woman has shot up dramatically. It is always funny to in a morbid way, to see a typical Spanish family from the non cities. Grandmother, is 5 feet if that, mother is a bit higher and then comes the grandchildren touching if not over 6 feet. All this in 3 generations.
Now this pattern has been repeated in every modern society over time. Birth rates in the rest of Europe have fallen in tact with more and more women joining the work force. The same has happened in the US. I would bet that the average age for a girl marrying in the 1800s in the US was around 15 to 17 and with in a year having a child.
And if we look at the Muslim population, the same thing is happening there.. although slowly. The Turks or other Muslims that arrived in Europe in the 1960s came from traditional families and brought their traditions with them. This meant girls were second rank citizens and expected to marry at a very very young age. By the time a girl was 18 she was basically expected to have at least one child if not more and of course be married.
Now the second and 3rd generation Muslims in Europe.. they do not follow that traditional form in the same way. They resist the forced marriages for one in a rising number, and that often costs their lives. And yes their birth rate is high compared to other Europeans, but considering the same trend as I mentioned above is slowly happening. Girls are against the wishes often of the traditionalist fathers being educated in basic schooling and even in higher education which was unheard off back home. Many of the women who came up during the 1960s are illiterate for example. But these changes in turn pushes Muslim womens weddings further out in the future relative speaking. So instead of getting married at 15, they now get married at 18+. This also means that they get children later and often far fewer than their own parents.
But to be brutally frank.. I would rather have a lower birth rate than go back to the "good old days" where women were a trade good.