We study the population of ultra-diffuse galaxies (UDGs) in a set of eight nearby (z < 0.035) galaxy clusters, from the Kapteyn IAC WEAVE INT Clusters Survey. We report the discovery of 442 UDG candidates in our eight field of views, with 247 of these galaxies lying at projected distances <1R200 from their host cluster. With the aim of testing theories about their formation, we study the scaling relations of UDGs comparing with different types of galaxies, finding that in the full parameter space they behave as dwarf galaxies and their colours do not seem to correlate with their effective radii. To investigate the influence of the environment on the evolution of UDGs, we analyse their structural properties as functions of the projected clustercentric distance and the mass of their host cluster. We find no systematic trends for the stellar mass nor effective radius as function of the projected distance. However, the fraction of blue UDGs seem to be lower towards the centre of clusters, and UDGs in the inner and outer regions of clusters have different Sérsic index and axial ratio distributions. Specifically, the axial ratio distributions of the outer and inner UDGs resemble the axial ratio distributions of, respectively, late-type dwarfs and dwarf ellipticals in the Fornax Cluster suggesting an environmentally driven evolution and another link between UDGs and dwarf galaxies. In general our results suggest strong similarities between UDGs and smaller dwarf galaxies in their structural parameters and their transformation within clusters.