What will happen to the steppe if all the ungulates living there are exterminated?

If you destroy all the ungulates inhabiting the steppe expanses and stop grazing livestock there, there will be no one to eat herbaceous plants, preventing their spread, therefore, the steppe will quickly turn into a forest-steppe massif overgrown with shrubs and rare trees, the seeds of which are carried by the wind. On the other hand, there is a possibility that the vegetation that has appeared will provoke frequent fires and this will lead to the emergence of a desert, since sands will begin to attack the empty space.