A study is presented of migration of optical or magnetic excitations on percolation clusters which terminates upon reaching a trapping site. The theory is based on the extension of results from the theory of random walks to systems without translational invariance, together with the use of scaling concepts. For the case of an excitation which resides on one type of atom in a randomly mixed crystal near the percolation threshold, new power laws for the time and concentration dependences of the mean number of sites visited at time t of the kinetics of arrival at traps are obtained. Some of these results are also tested for the first time by numerical simulations.