Someone I met this weekend told me about a feature in VirtualDJ he called "Beatgrinding". Some googling found me that it's actually called "beat gridding" and this explanation: http://www.virtualdj.com/forums/66481/Mix_lessons/Beatgrid_effect.html
Essentially it's a plugin that samples the 16 (presumably configurable) beats following when you activate it and throws them into a Hydrogen-style drum machine. The upshot is that you can very quickly on the fly cut and chop a section of a track and have it get looped underneath a different (or the same?) track.
I think we should (in the far future) add this since it looks like it would be really really useful. I get that similar separate apps already exist to do this, but it would be tricky to integrate them on the fly into a song with an unkown BPM.
Traktor already uses "beat gridding" to mean manually laying out a grid of beats, so I'm going to keep calling this grinding.
In general, this is called beat-slicing. There are a few programs that do it, including Propellerheads' ReCycle, and FreeCycle (which I think is defunct). OpenAV has a program called Slicr in the works that does it too. Not sure how that could integrate with Mixxx though.