I'm hoping people say "DSLR" when they actually mean "interchangeable lens system camera" because the best ILSC cameras for video are not Digital Single Lens Reflex design but are mirrorless. These include the Panasonic Lumix G series, Sony A7xx and A6000, and Samsung NX1 and NX500.
The biggest single issue against DSLRs is the limited shot length. Shooting a concert is a terrible time to have the camera stop at 20 or 30 minutes and require a restart. Whether it's file handling, sensor overheating, or simply a design choice, I know of no DSLR that will shoot until the card fills up or the battery runs out, while this capability is common in mirrorless cameras.
The big advantage of the Lumix cams is the smaller Micro 4/3 sensor. It's common to take the position that a bigger sensor is better, but like most things in life there are tradeoffs. A larger sensor gives a shallower depth of field for a given aperture which is great for portraits and dramatic shoots, but it makes the area in focus smaller and catching focus trickier. More importantly to me, the larger sensor requires larger heavier lenses for the same focal length and max aperture.
Fran