In Android 13, the gesture bar remains mostly static, except when you switch from an app to the Recents overview, the bar moves with the app card, but as soon as you release your finger, it returns to its place without any animation.
Thankfully, that’s about to change. In Android 14 Developer Preview 2, we managed to enable an animated navigation bar gesture pill. The spring animation plays when you bring up the Recents overview menu, when you swipe to home, and when you go from the app to the Recents overview (it no longer goes with the app card).
This is not the final look. At this stage, it’s still extremely unstable, because as you can see from the demo, the animated pill overlaps the static one. This will most likely be fixed in future builds.