Despite being made out of wood, the UI in DKCR has an amazing amount of "guish". Every element on screen bounces around like Jell-o, squashing on impact and stretching at high speeds. It works hand in hand with the speed of the effect - it's so blazingly fast that even solid wood starts bending.
The speed of the UI is borderline ludicrous - only 13 frames (0.433 seconds). It makes moving through menus incredibly brisk and easy; the time between pressing a button and seeing the results is under a second. The effect is so fast that several moves only get 1-2 frames of airtime, betraying the amount of underlying work that went into just one frame.
Unlike the Title UI, which has the elements make a hard stop at the end of their animation, the elements in this screen have more give. Each one of the elements over-extends, shooting past the desired position before settling into place. What's amazing is that this part of the animation even exists, since the over duration means the over-extension is only visible for a single frame.
The "New File" elements are the only elements the player can actually select. So, they get a lot more attention! The UI makes a 5 frame sound wave when it reached its destination, and even has speed lines following the trajectory.
While the effect is fast, the outro is even faster - elements leave the screen in only 9 frames (0.3 seconds), 4 frames less than the intro. When a player chooses to do something (like pressing a button on the UI), they've made a decision and want to see the results. So even if it's just a 4 frame difference, get out of their way ASAP!
Like the Title UI, this simulation relies almost entirely on springs. Springs help make the animation feel more physical.
It doesn't matter what part of the UI you are - you will squash and stretch, and you will like it. Every element onscreen does the squashing and stretching, even the button legend at the bottom of the screen.
While the springs are cool, there's a nagging suspicion this effect is actually hand authored. Given the number of people on the project, it makes more sense that they simply copied this effect to each part of the UI. Especially given the amount of tweaking required to get the spring physics to play nice.