Well, video became available on DSLRs when sensor readout speeds became quick enough. Canon has lead the pack here, first through readout speeds and actually having a real-time feed to the display (if you've seen older Sony and NIkon DSLRs / DSLTs / MILCs, you know what it means to not have this), while Sony has provided more recent competition and Nikon looks to actually be catching up.
And they have just about everything figured out. The lenses are actually the biggest problem as photographic lenses tend to be tuned more for acuity, whereas cinema lenses (or video lenses in general, meaning broadcast lenses) tend to eschew absolute acuity for other optical corrections as well as being parfocal (for zooms, which means that the focus point doesn't change when the zoom ratio changes) and having little to no focus breathing (which means that the field of view doesn't change when focus changes). There are other optical traits that tend to differ a bit as well, such as handling of flare and chromatic aberrations.
And to put it simply, a fast zoom lens that is both parfocal and exhibits no focus breathing is not a cheap endeavor, even if it has the optical quality of a glass bottle!
Beyond glass, things like rolling shutter producing a 'jello effect' due to slow readout speeds (or just lack of global shutter) and focus and stabilization inconsistencies are still being ironed out, but are so very close that it'd be hard to tell for most video use. Assuming the limitations are mitigated, we're at the point where being able to tell the difference between a 'cinema' or 'video' camera and a photography-focused MILC is very difficult.