What is required to succesfully drive a King into the corner is very difficult to answer. For one, as the 12x12 checkmates thread shows, it is dependent on the size of the board. When you solve the end-games by retrograde analysis, there are several possibilities:

Usually you start with very many checkmates, along the edge, none of them enforcible. (Unless you have something like a Queen). Pieces like Bison that cover only 2 orthogonally contiguous squares do not have those, however, and solely rely on a handful of corner checkmates. Usually all longer mates are ancestors of those corner mates.

Sometimes the longer mates die out immediately, like in the case of Gnu. There exist mate-in-1 positions, but no mate-in-2 positions, so afterthat, you are done. This is just an unlucky coincidence of the piece not being able to make the critical step between a position needed to force the bare King to step into a corner, and the square where it needs to be for a corner checkmate.

More often the number of longer mates increases very slowly with their duration, and then hovers for a long time around a very moderate number. These are the positions where the bare King is already trapped on the edge, and has to be driven into the corner with very precise play. On large boards, the number of longer mates even tends to decrease again, because it becomes easier and easier for the bare King to actually flee towards the corner voluntarily, the attacking pieces not being able to all follow it quickly enough to keep it trapped there, so that it can then escape along the other edge. So you have to confine the bare King more precicely as you are further away from the corner, not only cutting off its way back to the middle of the edges, but also preventing it gets too much of a headstart towards the corner. 

This leads to a decrease of the number of positions. On boards that are too big the number of mates actually decreases to zero before you retrogradely reached the middle of the edge, and the game is generally drawn: there is no way to drive the bare King towards the edge without it reaching the edge in the middle between corners, and there is no way to drive it over such a large distance to the corner without it escaping in one direction or the other.

If you survive (in retrograde time) until the middle of the edges, though, the number of longer mates suddenly starts to explode. It is usually very easy to drive a bare King to the edge, if you don't care where it will hit the edge. Unless the board is really big. But on 8x8, if your own King is in the center, the opponend is already driven onto the second rank. So in the early phase, from very unfavorable position, almost any sequence of moves that step your King plust its lieutenant towards the center (using opposition to drive the opponent out of it) is a direct route towards the checkmate, and there are very many possibilities for this. 

On all boards I have tried, (upto 16x16) once you reach the point where the number of longer checkmates starts increasing again, it usually fills up the entire space of positions. Either that, or it slowly peters out before it ever got big.