Climate change has an effect of course. But stopping land from turning into desert starts with giving vegetation a chance instead of hungry sheep destroying anything green. Some grazing is OK. But overgrazing destroys the land. Once the vegetation is gone, the soil erodes and it no longer absorbs what little rain comes down.
There are some land restore projects in the middle east where the primary action was simply keeping the sheep from eating anything green that sticks out above the soil with simple fences. That seems to work and can restore barren landscapes in a few short years. Some studies in e.g. Jordany and the UK (places like Scottland should be covered in atlantic rainforest instead of being grazed into a barren landscape) and elsewhere seem to indicate that keeping sheep away for a while gives plants and trees a chance to re-establish themselves.
Trees are really vulnerable in their first few years and a tasty snack for grazing animals that without natural predators can strip the land of anything green in no time.
Overgrazing can be a problem, but undergrazing can be just as big of one.
Healthy pasture requires a certain rhythm/ amount of hoof traffic to stay healthy.
It's why land restoration in the (US) Midwest/West tends to do much better if it includes a reintroduced (managed) grazing component.
And why even wild pasture in Africa typically has a cycle of trample and/or natural burn as part of it's life cycle.
This may or may not apply to previously forested land, depending on what's in-situ, but grazing should be seen just as much as a positive requirement, as overgrazing is seen as a detriment/negative.
Now if your goal is reforestation instead of just healthy pasture or other sustainable ecotype, that's different .
But don't assume just because land can sustain forest, that forest is the 'natural' ecosystem. See: the US history of pasture vs forest. There's more forest now than there was pre-euro settlement.
Much like exercise and muscles or immune systems and exposure. Life thrives on just the right amount of tension or stress . Sedentary is equivalent to tomb.