The only places I've experienced lag in 8.1 dp on my 820 are when certain apps are loading. The games hub is the biggest offender to me. It seems to load, reload, and then load some more. What a bunch of garage. I guess they really don't want people to use it. Compare it to 8.0 where it just opened and worked.
8.1 dp on my 920 is a different story. Start screen animations are not fluid. The games hub, menus in the camera app, and several other places stutter noticeably. Everything seems to be running at 30fps instead of 60fps like it should. Hopefully this will be fixed when 8.1 Cyan rolls out (yeah, right).
That lock screen app is a ****load of ****.