Solute transport by river and stream flows in natural environment has significant implication on water quality and the transport process is full of uncertainties. In this study, a stochastic one-dimensional solute transport model under uncertain open-channel flow conditions is developed. The proposed solute transport model is developed by upscaling the stochastic partial differential equations through their one-to-one correspondence to the nonlocal Lagrangian-Eulerian Fokker-Planck equations. The resulting Fokker-Planck equation is a linear and deterministic differential equation, and this equation can provide a comprehensive probabilistic description of the spatiotemporal evolutionary probability distribution of the underlying solute transport process by one single numerical realization, rather than requiring thousands of simulations in the Monte Carlo simulation. Consequently, the ensemble behavior of the solute transport process can also be obtained based on the probability distribution. To illustrate the capabilities of the proposed stochastic solute transport model, various steady and unsteady uncertain flow conditions are applied. The Monte Carlo simulation with stochastic Saint-Venant flow and solute transport model is used to provide the stochastic flow field for the solute transport process, and further to validate the numerical solute transport results provided by the derived Fokker-Planck equations. The comparison of the numerical results by the Monte Carlo simulation and the Fokker-Planck equation approach indicated that the proposed model can adequately characterize the ensemble behavior of the solute transport process under uncertain flow conditions via the evolutionary probability distribution in space and time of the transport process.