The Great Wall didn't really prevent people from crossing into China, it was more of an early detection system than a true barrier. The Steppe pastures that horse nomads want extend westward, so it was a natural channel for them to go that way.

Pastoralists like the Scythians had been making regular crossings into Europe through the Danubefor centuries before the Great Wall.

The real impetus for crossing into China, and thus the reason for the Great Wall, was to take over parts of a rich civilization. That was a regular occurrence even with the Great Wall in place. When Steppe conquerors noticed new goods being traded from a rich civilization at the other end of the world like Rome, they migrated in that direction.