What noises can be sliced? A well-known property of Perlin noise (with FBM) is that if you take a slice of it, the result is still Perlin noise.  So, by taking successive slices of 4D  noise, you get smoothly shifting 3D noise, like a cloud or a lava lamp.

What other noises have this property?
Worley noise, also known as cellular noise, has the same property. It just as easily implemented as Perlin noise and easily extends to higher dimensions. Thus the slicing of 4D Worley noise will produce a 3D Worley noise. However, it is not necessarily a noise function but rather a texture function, producing cellular-like characteristics. With FBM applied to Worley noise some interesting patterns can be created.

You can find additional types of procedural noise functions in this survey paper.