Voxel graphics for games is mostly restricted to non-deforming bodies. Games like Comanche and Outcast made use of an optimized variant of voxel rendering for their terrains with great results in the 90s. But I have yet to see an animated face in realtime rendered using voxels or points. I would love to see that done, but this is a hard problem.
If you're doing it in a pure software implementation and with no tilted camera, you could optimize this heavily. You start at the bottom of the screen and for each vertical row of pixels you march along the heightmap in 2D and project the texels to the screen. You also remember the current hightest Y coordinate that you've drawn. If the new texel is projected higher than that, you draw another vertical line to that position with the texel color and remember that position. So you're filling up your screen from bottom to top in an extremely efficient way.
edit: found the video (I find myself skipping the teaser quite often these days). They show recordings, not animations. This may seem like splitting hairs, but it's not. An animated rig of some kind can be adjusted through parameters that are entered by an animator to create or alter the movement. This is also the basis for technology like animation blending and inverse kinematics which (in a nutshell) are some of the techniques that make movement in current games so plausible. If this is just a static stream of volumes that is played back like a flipbook, then no easy adjustments are possible.