A new kinetic theory of electron scavenging controlled by non-classical electron transport in disordered solids has been developed. This theory is based on the percolation approach to the description of a trap-to-trap tunnelling motion of electrons in solid systems with a bond disorder. Such an approach is applied to determine the concentration of trapped electrons as a function of time for a disordered medium containing randomly placed, immobile scavengers. Two limits of direct physical significance have been considered in detail, i.e. the cases when the concentration of trapped electrons is either much less or equal to that of scavenging molecules. The comparison between present results and those obtained by other authors is discussed.