A lot of levels really intend the player to outrun the monsters and having them doggedly chase down the knight across time and space would be terrifying. The effect this would have on arenas would also be quite profound, mostly on enemies like mecha knights and those respwning crusaders that already chase you down- imagine if mecha knights were as fast as grievers... the sheer amount of rage would be unbelievable. I'm going to have to say that this is probably not a good idea, considering how poor many players are at the game- like I seen it and woah boy this would crush their souls.

I'm also going to have to say that this sounds awesome and challenging for more advanced players, and that I'd love to have T3 levels based around truly awful monsters that hunt you down.. or a mechanic that works similarly, like if they made some sort of energy field that boosted monsters speed without effecting the knight directly. This field could then be applied to large parts of the area, or even potentially reversed by a switch or as an item effect.. which would be like stagger storm, but would instead just slow enemies while they're inside of it. OK maybe too close to stagger.

Another idea would be an enemy that grants other enemies a speed buff. Like if they made more silkwing variants that instead of healing bestowed speed up, defense up, attack up, what have you. So you have some level where the knight is running through, and speedy silkwings + mecha knights spawn.. all of a sudden, running down that hallway to escape doesn't seem like such a grand idea.

And finally you could just directly take enemies and do what you're saying, pinning it as a separate challenge area. I'd be down with that.