First, and this might sound pedantic, but it's worth specifying that we're talking about Turkic peoples, not "the Turks" when we talk about the Seljuqs or the Ghaznavids or the Zengids or whatever.

Secondly, and in more direct answer to question, I don't think it should be viewed necessarily as a zero sum relationship, at least not at first. The reason why Turkic peoples became so prominent in the heartlands of Islamdom is because, at least initially, they were a militarily powerful group of Muslims on the near-periphery of Abbasid territory and were a useful component of the caliphs in assembling a military force. This developed into early forms of a slave soldier or mamluk system, and the result was that Turks took on a powerful military role within the empire as part of the Caliphs themselves maintaining power.

But there's two caveats to this. One is that very early in Abbasid rule you start to see a splintering of the Islamic empire, so Ummayad Spain breaks away for instance. Clearly there are opportunities for enterprising and dynamic tribal peoples to dominate specific regions of Islamdom and there's little Baghdad can do about it.

So Abbasid decline sees not just the rise of Turks but other local dynasties as well, so you have the Tulunids in Egypt (Turkic), the Ghurdis (Persian), the Samanids (Persian), the Buyids (Persian) and other regional powers who are, either temporarily or with some permanence, able to break away from and challenge Abbasid control.

Eventually it's the Buyids, an Eastern Persian dynasty, that is first able to conquer Baghdad and reduce the Caliphs to a position of formalized political subservience in the 10th century, but their powers had been significantly reduced by their own Turkic slave soldiers before hand, particularly following the Anarchy at Samarra.

Even the Buyids, however, had to use Turkic horse forces because their own troops were strictly infantry, along with a mix of Kurds and Arabs.

But I think if I would point to a couple things that gave them regional influence that led to the rise of these Turkic dynasties it would be 1. regionalization that allowed for the rise of independent power bases, 2. the institutions of slave soldiery, where masses of Turkic tribesmen were brought in, at first to support the regimes, and then practically inevitably to dominate them from within. Finding ways to use slave soldiers but prevent them from taking over the state in hereditary fashion is, as I understand it, part of what led to the more sophisticated systems of the late Egyptian Mamluks and also the Ottoman Turkish devshirme. But that's all post-Abbasid caliphate.

I always tend to point to Marshall Hodgson's The Venture of Islam as being a good, sweeping overview of these dynamics that covers many of the social elements involved as well.