We considered diffusion driven processes on small-world networks with distancedependent random links. The study of diffusion on such networks is motivatedby transport on randomly folded polymer chains, synchronization problems intask-completion networks, and gradient driven transport on networks. Changing theparameters of the distance dependence, we found a rich phase diagram, with differenttransient and recurrent phases in the context of random walks on networks (orwith different smooth and rough phases in the context of noisy task-completionlandscapes). We performed the calculations in two limiting cases: in the case of rapidnetwork update, where the rearrangement of the random links is fast, and in thequenched case, where the link rearrangement is slow compared to the motion of therandom walker or the surface. It has been well established that in a large class ofinteracting systems, adding an arbitrarily small density of, possibly long range,quenched random links to a regular lattice interaction topology will give rise tomean-field-like behaviour (i.e., the small-world-like random links can be treated in amean-field fashion). In some cases, however, mean-field scaling breaks down in‘low dimensional’ small-world networks, where random links are added to a lowdimensional regular structure. Examples of such cases are the common diffusion and theEdwards–Wilkinson process, the main subjects of this paper. This breakdown can beunderstood by treating the random links perturbatively, where the mean-fieldprediction appears as the lowest order term of a naive perturbation expansion. Theasymptotic analytic results are also confirmed numerically by employing exactnumerical diagonalization of the network Laplacian. Further, we construct a finite sizescaling framework for the relevant observables, capturing the crossover behavioursin finite networks. This work provides a detailed account of the self-consistentperturbative and renormalization approaches briefly introduced in our earlier work.