Hair dramatically sways on the first frame of new position

To get rid of the sway, you can add a second at the start of the scene for the hair etc to settle down. When you teleport the puppet, add another second of things not moving again there. Then you can then trim off the start and middle of the scene using Premier Pro.

Alternatively you can create two scenes and each scene can have a "workspace" set. If you set that to start a second into the scene, then when you export the work area it only includes what is in that range. This lets the physics engine settle down any bouncing from dangles. 

With two scenes I am not sure if you can create a new third scene and put the two real scenes in it and do a single export... but what I do is export the scenes separately and use Premier Pro to put them together. Since I could have 10 or 20 scenes in a single video, this also allows me to export just the scene with changes. This makes it faster when doing final tuning and edits. I don't export a single long video, I re-export only the scene where I made a change.

So there are a few ways, but yes - there can be bounce from dangles. This introduction of the workspace feature was to help with this, allowing you to specify per scene to start a second or two into the scene, giving physicals a chance to find equilibrium. E.g. with long hair and strong springiness, it can take a few seconds to bounce up and down before it gets stable.