What forces cause the eluting solution to move along the chromatography paper?

The eluent moves along the chromatography paper because of capillary action.

The molecules of eluent have strong attractions (adhesive forces) to those of the paper.

The molecules of eluent also have strong attractions (cohesive forces) to each other.

Thus, as the adhesive forces pull eluent molecules in contact with the paper along its surface, cohesive forces pull the rest of the molecules along with them.