Time-nonlocal transport models can describe non-Fickian diffusion observed in geological media, but the physical meaning of parameters can be ambiguous, and most applications are limited to curve-fitting. This study explores methods for predicting the parameters of a temporally tempered Lévy motion (TTLM) model for transient sub-diffusion in mobile–immobile like alluvial settings represented by high-resolution hydrofacies models. The TTLM model is a concise multi-rate mass transfer (MRMT) model that describes a linear mass transfer process where the transfer kinetics and late-time transport behavior are controlled by properties of the host medium, especially the immobile domain. The intrinsic connection between the MRMT and TTLM models helps to estimate the main time-nonlocal parameters in the TTLM model (which are the time scale index, the capacity coefficient, and the truncation parameter) either semi-analytically or empirically from the measurable aquifer properties. Further applications show that the TTLM model captures the observed solute snapshots, the breakthrough curves, and the spatial moments of plumes up to the fourth order. Most importantly, the a priori estimation of the time-nonlocal parameters outside of any breakthrough fitting procedure provides a reliable “blind” prediction of the late-time dynamics of subdiffusion observed in a spectrum of alluvial settings. Predictability of the time-nonlocal parameters may be due to the fact that the late-time subdiffusion is not affected by the exact location of each immobile zone, but rather is controlled by the time spent in immobile blocks surrounding the pathway of solute particles. Results also show that the effective dispersion coefficient has to be fitted due to the scale effect of transport, and the mean velocity can differ from local measurements or volume averages. The link between medium heterogeneity and time-nonlocal parameters will help to improve model predictability for non-Fickian transport in alluvial settings.