I am not at computer, but I think from your video the problem is that this how the walk behavior works. Puppet keeps walking while walk is enabled, not in sync with playhead. It is confusing. Physics is the same.
If you click the refresh scene button it gets reset to the original puppet position.
note - this is a guess from my phone (very small screen)
So I downloaded the project, and yes, I think its the problem above but I will expand a bit more. And yes, its confusing.
The walk behavior has a "body speed" setting which means "while the body is walking, move it by this rate as well." Zero means "move the legs but don't move the puppet". You have it at 24% which means "when doing the walk movements, physically move the body as well."
In addition you put a dragger on the duck while walking.
The walk behavior will move the puppet while the walk behavior is "active". This could be playing back a previous recording. But if the red armed button is on the property of the puppet, then it will not play back the current value (so you can stop the Walk behavior by arming it). If you record new values it suppresses the old values.
Dragging the play back head around the timeline sometimes resets the object to its start location, other times you are on your own. But the most important button to remember is the reset button puts the puppet back to its starting position.
How to reduce confusion? You can set the body speed to zero and use position X blends instead. That stops the puppet working off scene. Then Position X/Y etc can be used to move the puppet more reliably.
Alan is correct. It's confusing and frustrating. I have solved this problem somewhat because when I split characters, the behaviors for the puppet begin over again. I do this because I often split a character when I change lyrics in a song. It helps me because I don't have to go back to the beginning to record and edit. As you can see if you read the post about how to organize files, there is more than one way to do this.