The tail is "jerky" when the motor looses power for that short millisecond or so. I think this is normal as your gyro is holding the tail by keeping the tail rotor blades at a certain degree angle. When the power is lost in that millisecond...the gyro has to put more pitch into the tail to keep a relative heading. So that would to me look to be normal...I may be wrong though.

I would guess that its either the motor itself, or your ESC. Do you have spares of both to try swapping out to see if it cures your issue? Id say change the ESC out first since its the easiest of the two.

I would guess that the reason it takes a minute or so to start happening is because its a heat related issue with either the motor or ESC. Doesnt do it till it gets hot enough, makes a bunch of sense to me.