I am using Blender version 2.93.1 on a Windows 10 PC.
I made three globes. One as a sphere and two as cubes. The sphere and the first cube have an "earth" texture, like a globe. The second cube is more like a polygonal globe with a lot of "extruded" areas and solid areas.
All three Objects in the same position in the room. I scaled the two cubes small and only the ball is large. At the beginning of the animation the sphere shrinks and cube number 1 grows to the original size of the sphere. The first cube is followed by the second at a short distance. The first cube shrinks, making way for the second cube. That should act like a transformation. Now only the second cube can be seen. This has areas that are combined to form a vertex group. These in turn are the basis for a particle system (hair). They take the form of "cities" and "trees". Now I have learned and experienced that I can animate the transformation from sphere to cube with the "Cast" modifier. So I hidden the sphere and the first cube and applied the "Cast" modifier to cube number 2.
Now I have the problem that the "cities" and "trees" do not stick to the surface of the transformed cube and float in the air. Exactly where the surface of the cube would normally be. How can I realize that the objects in the particle system move along with the surface changed by the "Cast" modifier?