Optimizing excitation transport in quantum networks is an important precur-sor to the development of highly-efficient light-harvesting devices. We investigate thephenomenon of dephasing-assisted leakage from trapped states, which may ensure effi-cient transport of excitations across a network. We consider three small networks withknown trapped states and study the effect of Markovian quantum noise and classicalnoise sources with different correlation times, from very low frequency non-Markoviannoise to white noise. We show that the excitation - otherwise stalled in trapped states -is able to diffuse rapidly through the networks from a source to a sink, with efficiencybeing generally maximized for intermediate correlation times.