Performing atomic-scale simulations, we study the interaction of the scanning tunneling microscopy (STM) tip with mesoscopic islands at zero bias voltage. Our calculations reveal tip-induced shape transitions in Co islands on Cu(100) as the tip approaches the surface. The structure of the islands and of the tip are found to depend strongly on the tip-substrate distance. A significant influence of the tip on atomic diffusion on the top and at the edges of the islands is demonstrated. The size-dependent strain relief in the islands caused by the tip and by the substrate is found to play a key role in atomistic processes on islands. Our results show that for certain tip-surface separations the hopping diffusion of Co adatoms on the top of Co islands and the upward mass transport at the edge of the islands can be strongly enhanced. Our findings point out the possibility of manipulating atomic motion on mesoscopic islands using the STM tip.