Chain made out of 2D Hinge Joints isn't moving around, also it stretches weirdly

I've been using Unity to make a little platformer and it's been great so far! I've just been having one issue. I want to create a physics-based cape on my character, using a chain of Hinge Join 2D's connected to each other. Basically what they're doing in this video: https://youtu.be/vRPW3H4X9PE?t=299 with the character's ponytail.

The concept of it works on its own, so it reacts to gravity & changes in rotation like you'd expect it to. But when I attach it to my character, it has a couple problems:
1. The cape sticks exactly with my character when it moves. It doesn't flow behind him when he walks forward, or stay up for a second when he jumps down, or anything, just stays exactly where the character is, in a straight line up and down. You can see this in a video of it: https://www.youtube.com/watch?v=0KNbz4ZjMfw
2. The pieces of the cape slowly drift apart up and down, also shown in the video. I'm not sure why this would happen.
My setup looks like this: Essentially the cape is a child of the player object, and then each piece of it is connected to the piece above. I can give more info (or you can look in the video) if necessary.