The North "won" the resulting battle of competition, in that far more creatures successfully moved south than travelled in the opposite direction. These were, on the whole, creatures we're familiar with across the Northern Hemisphere, because the proximity of North America to Asia had already led to some blurring between the animal groups there. So, it's as a result of this, the Great American Interchange, that South America has foxes, deer, cats (most obviously jaguars and ocelots), mice, rats, weasels, otters, and rabbits.
Not only that, but some of the animals that headed south later died out in their northern homeland, so that today, we think of them as being uniquely or primarily "South American". These animals include the llamas and the tapirs, and, to a lesser extent, the peccaries or javelinas, which still survive in Mexico and the American south.