Flow and transport in coupled channel-matrix systems are ubiquitous to many environmental and engineering applications such as flows in fractured porous media over canopies and in membrane filtration units. The multiscale nature of such systems, where the horizontal length scale is often orders of magnitude larger than the vertical one, allows one to employ vertically averaged descriptions of the system. As a result, two-dimensional transport in the channel and the matrix can be upscaled to a coupled system of transient one-dimensional advection-dispersion equations, where matrix and channel properties can be analytically related to macroscopic transport observables. In this work, we first develop a semianalytical solution based on integral transforms that can be employed to predict macroscopic transport in channel-matrix shear flows in a computationally efficient manner. Then we demonstrate that under appropriate dynamic conditions, the coupled system at the macroscale can be further simplified to a single upscaled one-dimensional advection-dispersion equation, which admits an analytical closed-form solution, thus enabling real-time macroscale concentration estimates in relevant applications.