First of all, make sure the positions of your rig are correct BEFORE the animation actually plays. Create keyframes at the beginning of the track so the animation will default to that position first.
I think the solution to your problem would be to adjust the timing of your animation- specifically using “empty” keyframes, which basically pause the animation for a period of time. You just copy the previous keyframe and move it over. That way the animation will stay completely still when that keyframe plays because it isn’t changing anything.
The first example is probably what you have. The animation immediately begins playing after the first keyframe.
In the second example, the animation doesn’t play until the second keyframe is reached, because the first two keyframes are identical and nothing changes. The animation here will also play faster because the keyframes with motion are closer together.
Just keep tweaking around with the keyframe timing until it fits.
