I recall reading this somewhere: That the "great" civilizations in the middle east gradually moved further and further west, as each civilization over-farmed it's area (and I guess caused desertification?). In a broad sense is this true?
It certainly isn't true in the history of the past 1500 years.
The capitals of the Caliphate moved north from Medina to Kufa to Damascus. Then jumped east to Baghdad. North again to Samarra. Then after the anarchy at Samarra while the Caliph remained in Baghdad the real political capital of the Seljuks moved around eastern Persia before settling in far eastern Persia in Merv, modern Turkmenistan before the arrival of the Mongols. Following the Mongols the Timurid rulers of Persia kept moving east, establishing their Capitals at Samarkand in modern Uzbekistan and then at Herat in modern Afghanistan.
So moving west in response to desertification? Not in the period I've just described from about 600-1500.
That's not to say desertification wasn't an issue. There's a good deal of debate about whether and why Iraq and parts of Persia seem to have been abandoned at various points in time. Right before the Islamic conquests there's evidence that large rural populations up and left Iraq, but it's not really clear why. Likewise the Mongols have been accused of ruining (intentionally? that's the debate) through mismanagement the Sawad, the black plains of Iraq.
But I haven't read anything about the effects of this being a push west. If anything, as I've described, they pushed east.
edit: typo
In "Collapse" Jared Diamond argues that Indo-European "centers of civilization" moved westward from the fertile crescent to the Aegean to the Adriatic as natural resources (trees, fertile land, wild animals, etc.) were depleted. Of course, this is a Western-centric argument, ignoring the Indus valley civilization and the Eastern Holy Roman Empire. There is also an inherent questionability in Diamond's work; I only bring it up to give context.