The evolution of the first five nonnegative integer‐order spatial moments (corresponding to the mass, mean, variance, skewness, and kurtosis) are investigated systematically for spatiotemporal nonlocal, fractional dispersion. Three commonly used fractional‐order transport equations, including the time fractional advection‐dispersion equation (Time‐FADE), the fractal mobile‐immobile (MIM) equation, and the fully fractional advection‐dispersion equation (FFADE), are considered. Analytical solutions verify our numerical results and reveal the anomalous evolution of the moments. Following Adams and Gelhar's (1992) work on the classical ADE, we find that a simultaneous analysis of all moments is critical in discriminating between different nonlocal models. The evolution of dispersion among the subdiffusive to superdiffusive rates is then further explored numerically by a non‐Markovian random walk particle‐tracking method that can be used for any heterogeneous boundary or initial value problem in three dimensions. Both the analytical and the numerical results also show the similarity (at the early time) and the difference (at the late time) of moment growth for solutes in different phases (mobile versus total) described by the MIM models. Further simulations of the 1‐D bromide snapshots measured at the MADE experiments, using all three models with parameters fitted by the observed zeroth to fourth moments, indicate that (1) both the time and space nonlocality strongly affect the solute transport at the MADE site, (2) all five spatial moments should be considered in transport model selection and calibration because those up to the variance cannot effectively discriminate between nonlocal models, and (3) the log concentration should be used when evaluating the plume leading edge and the effects of space nonlocality.