What you're asking for should be pretty straightforward. I can only really invision three collision cases:
- sphere with side
- sphere with edge
- sphere with corner
The first is very similar to testing for AABB collisions, and the last is almost identical to sphere-sphere. The second is a little trickier I guess, I'd be projecting to a plane to which the edge is normal (for AABBs that's effectively just ignoring a co-ordinate) to effectively turn it into a 2D circle-point problem.
If you provide a little more detail on motion (obviously the nature of motion of objects involved is typically very important in a priori), I might be able to give a better response.