Here, working in Object Space. The Position output of the Voronoi node is the location of the feature point from which the attributes (distance, etc) of the local cell are measured. If you subtract it from the shading point, you get the vector from the feature point to the shading point .. a mini UV map for each cell, with its origin at the feature point.
If you Nornalize those coordinates, you get a measure of direction from the cell origin expressed as XY (all the points in the same direction from the cell origin will all have the same unit vector), so that can be used for radial effects. Here, the vector is used to look up into a Noise texture.
Edit: Working in 3D, and normalizing at every stage, effectively ensures all the points used in the direction calculations are projected onto the unit sphere, towards or away from its center:
..and so straight lines become arcs:
(but this will work only on a sphere, not on any curved surface, and, to work in Object Space, the origin of your sphere-object should be at its center)
You can animate the effect by scaling the sphere onto which the feature-point is projected:
With this sort of result:
This is the simplest way. To get more specific effects, you may have to extract, say, the Normal Z to work with, which would be a bigger tree.