We considered diffusion driven processes on power-law small-world networks: random walk on randomly folded polymers and surface growth related to synchronization problems. We found a rich phase diagram, with different transient and recurrent phases. The calculations were done in two limiting cases: the annealed case, when the rearrangement of the random links is fast (the configuration of the polymer changes fast) and the quenched case, when the link rearrangement is slow (the polymer configuration is static) compared to the motion of the random walker. In the quenched case, the random links introduced in small-world networks often lead to mean-filed coupling (i.e., the random links can be treated in an annealed fashion) but in some systems mean-field predictions break down, such as for diffusion in one dimension. This break-down can be understood treating the random links perturbatively where the mean field prediction appears as the lowest order term of a naive perturbation expansion. Our results were obtained using self-consisten perturbation theory. Numerical results will also be shown as a confirmation of the theory.