  Quantum transport in a one-dimensional (1D) quasiperiodic lattice with
mobility edges is explored. We first investigate the adiabatic pumping between
left and right edge modes by resorting to two edge-bulk-edge channels and
demonstrate that the success or failure of the adiabatic pumping depends on
whether the corresponding bulk subchannel undergoes a
localization-delocalization transition. Compared with the paradigmatic
Aubry-Andr\'e (AA) model, the introduction of mobility edges triggers an
opposite outcome for successful pumping in the two channels, showing a
discrepancy of critical condition, and facilitates the robustness of the
adiabatic pumping against quasidisorder. We also consider the transfer between
excitations at both boundaries of the lattice and an anomalous phenomenon
characterized by the enhanced quasidisorder contributing to the excitation
transfer is found. Furthermore, there exists a parametric regime where a
nonreciprocal effect emerges in the presence of mobility edges, which leads to
a unidirectional transport for the excitation transfer and enables potential
applications in the engineering of quantum diodes.
