Precise modeling of surface water quality requires transport theory consistent with observed data. Efforts to achieve model fits to river tracer test breakthrough curves (BTCs) have led to different approaches to the one-dimensional modeling of river corridor transport at the macroscale. The memory function model form practically generalizes most current approaches including the transient storage (TSM), multirate mass transfer, decoupled continuous time random walk, and time-fractional advection dispersion models. An alternative formulation of the memory function approach is the phase exposure-dependent exchange (PhEDEx) model, which is the TSM model that includes residence-time dependence of the rate coefficient for mass exchange from the hyporheic zone to the river. Using temporal moment analysis and numerical simulations, we explore the general consistency of the PhEDEx model with observed BTC data. With this approach the moments of the memory function can be estimated without requiring any prior information about the memory function form. We also examine means of estimating river/transport process parameters using the results of temporal moments analysis. The results show the failure of the PhEDEx model, and therefore the memory function approach, in capturing in particular the observed constancy of the BTC coefficient of skewness (CSK). The temporal moments equations can be applied as a useful tool to estimate some river/tracer test parameters such as the ratio of HZ cross-sectional area to the main flow cross-sectional area, which are time/cost consuming to measure. The applications of the analyses are demonstrated with some case studies.