Ableton Live is a great tool for Live remixing - just ask Ableton Artist Kutski. Check out this video, in which he cuts up Shy FX's jungle classic, "Original Nuttah", with an APC40 and Live:
Kutski performs a real-time remix in Live, shows you how
Curious to try this kind of live remixing out for yourself? Kutski has also produced this helpful tutorial video, in which he demonstrates how to work with cut up breaks and other pieces, and how to use Clip Follow Actions: