Check this out. It's a lecture by Allan Savory sponsored by Feasta, an Irish organization ("The foundation for the economics of sustainability"). There's a 10 minute version and a 1 hour version. The full version includes a lot of seriously interesting stuff not in the condensed version.
I'd love to have the opportunity to talk with Savory, or someone who knows his perspective well. There are some seeming discrepancies and over-reaches in his presentation, but he may have perfectly good answers to explain those.
For example, if you watch the full version, you'll see that he identifies "partial rest" (that is, allowing a small number of grazing animals to wander freely over a large plot of land) as being as destructive to the soil as "full rest" (no grazing animals at all), so that both result in desertification. He's got lots of photos and case examples of this happening in arid climates. But in his lecture, given in Ireland, he implies that the Irish are likely to see similar results since the standard grazing method there is to use partial rest. But separately, he explains that the biological cycle, as it relates to soil health, is quite different from arid zones to year-round-humid zones, like Ireland. So is Ireland at risk of desertification if the current grazing pattern is maintained for sufficiently long? Or is there a fundamental difference that means partial rest grazing is harmless in humid zones?
Update: I've just come across another video on restoring degraded land, "Hope in a Changing Climate." Curiously, in this case the proffered explanation for success includes reduction of grazing. That's not a direct contradiction of Savory's thesis–he doesn't deny that grazing can degrade soil, only that it inevitably will. Also, Savory is focused on grassland ecosystems while the locations in this video look to be, historically, hillside woodland.