Socially conservative societies will tend towards demographic growth, while socially liberal societies will tend towards demographic decline. As a result, over time, the pool of immigrants will be more socially conservative every generation. We are witnessing, more or less, demographic transformation occuring in Israel and Turkey right now as the more socially conservative religious elements reproduce faster than secular folk.
On life, the pro-life either win by winning, or win by losing. A research study in Australian demonstrated that genetics plays a role in shaping ones attitude towards abortion. Those with genes more favorable to abortion will, eventually, abort themselves out of the gene pool. As long as abortion is legal, the genetic propensity towards being pro-life will increase every generation.
Your hypothesis relies on Darwinian natural selection to be true, you know.