The task of finding an element in an unstructured database is known as spatial search and can be expressed as a quantum walk evolution on a graph. In this article, we modify the usual search problem by adding an extra trapping vertex to the graph, which is only connected to the target element. We study the transfer efficiency of the walker to a trapping site, using the search problem as a case study. Thus, our model offers no computational advantage for the search problem, but focuses on information transport in an open environment with a search Hamiltonian. The walker evolution is a mix between classical and quantum walk search dynamics. The balance between unitary and non-unitary dynamics is tuned with a parameter, and we numerically show that depending on the graph topology and the connectivity of the target element, this hybrid approach can outperform a purely classical or quantum evolution for reaching the trapping site. We show that this behavior is only observed in the presence of an extra trapping site, and that depending on the topology and a tunable parameter controlling the strength of the oracle, a hybrid regime composed of 90% coherent dynamics can lead to either the highest or worst transfer efficiency to the trapping site. We also relate the performance of an hybrid regime to the entropy's decay rate. As the introduction of non-unitary operations may be considered as noise, we interpret this phenomena as a noisy-assisted quantum evolution.