Particles offset animation by curved object

I have a truble:

This is half-sphere. On the sphere I did place a few little spheres by particles emitter.
Now I need to animate offset of that little spheres. For example when I make Z offset - all little spheres goes down by face of half-sphere. They repeat half-sphere surface.

When little spheres moved to the side of sphere - they gone. And new one rise from opposite side randomly.
Same like texture offset, but for little spheres.

The trouble is: I can’t find particles offset and scale (like mapping node).

Thank you!