Miner can and will, if he can, dig from a side, but at rare occasions they can dig also from under themselves and then the miner will fall 1 level down.
When you have 10 miners on channeling same area, then surely they will happily dig ground from under each other, which will result in miner's 1 level fall.
When dwarf falls, he can fall right down or... jump aside 1 tile and fall 1 level down from there. So, on steep edges of channeled plain make sure to channel first a line of ramps on the area's sides. Fall from 1 level is less damaging then fall from 5 levels.
As falling seems unavoidable in a long run channeling, I suggest never to channel multiple levels on top of each other at once. Channel next level only after level above has been entirely channeled first.
Injury will not happen often. In majority cases the miner will get just stunned for a short while. However when injury happens in the end, then in majority cases unlucky miner will lose some teeth from the fall. However small injury to little finger, which cannot be diagnosed, washed & bandaged and may (50% chance?) cause during next season a deadly infection, which will surely kill the miner in yet another season, is not unheard about either, though very rare like 1 in 1 million. If you channel a lot, like I do, when flattening map's surface, then you will see how it works for sure.