What's the reason of the oscillation in the brightness of lightning flashes? In this video, which shows a lighting flash caught on a high-speed camera, you can see that (after 21 seconds video) the brightness of the lightning flash is oscillating a few times.
Why do lightning flashes oscillate?
Is it because the electrons (all going through the same channel) in the flash, go up and down and every time "shoot through" to cause a negative charge on the opposite side of the flash, after which the process reverses, again reverses, and again,..., until the oscillation is over, like a dampened oscillation?
 A: 
Is it because the electrons (all going through the same channel) in the flash, go up and down and every time "shoot through" to cause a negative charge on the opposite side of the flash,

Your model of how lightning happens is wrong.
If you mean the variation in light intensity  on the upper left corner at about 21''

it can be explained by the mechanism of lightning.  The light we see is a great current of positive charges  moving one way and negative the other in the same path,

Lightning is a sudden electrostatic discharge that occurs during an electrical storm. This discharge occurs between electrically charged regions of a cloud (called intra-cloud lightning or IC), between two clouds (CC lightning), or between a cloud and the ground (CG lightning). The charged regions in the atmosphere temporarily equalize themselves through this discharge referred to as a strike if it hits an object on the ground, and a flash, if it occurs within a cloud. Lightning causes light in the form of plasma, and sound in the form of thunder. 

In effect a plasma is formed in that region. The variation in time is the variation in the available plasma , i.e. the number of positive and negative charges in the small volume of the path of the discharge. Variations can happen because the available energy from cloud and ground has variations, a cloud is discharged but gets charged again by a neighboring cloud, for example. The specific boundary conditions have to be known .
My guess for the enhancement and depletion on the left upper corner is that after the first flash the region became ionized and a loop path was a possible by-road for the energy trying to ground itself or go to the cloud, the depletion and enhancements happening because of local variations/bottlenecks as the charges discharge. One would need more information to be more definite.
