Stochastic subsurface transport theories either disregard local dispersion or take it to be constant. We offer an alternative Eulerian-Lagrangian formalism to account for both local dispersion and first-order mass removal (due to radioactive decay or biodegradation). It rests on a decomposition of the velocity V into a field-scale component V-omega, which is defined on the scale of measurement support omega, and a zero mean sub-field-scale component V-s, which fluctuates randomly on scales smaller than omega. Without loss of generality, we work formally with unconditional statistics of V-s and conditional statistics of V-omega. We then require that, within this (or other selected) working framework, V-s and V-omega be mutually uncorrelated. This holds whenever the correlation scale of V-omega is large in comparison to that of V-s. The formalism leads to an integro-differential equation for the conditional mean total concentration (c)(omega) which includes two dispersion terms, one field-scale and one sub-held-scale. It also leads to explicit expressions for conditional second moments of concentration (c＇c＇)(omega). We solve the former, and evaluate the latter, for mildly fluctuating V-omega by means of an analytical-numerical method developed earlier by Zhang and Neuman. We present results in two-dimensional flow fields of unconditional (prior) mean uniform V-omega. These show that the relative effect of local dispersion on first and second moments of concentration dies out locally as the corresponding dispersion tensor tends to zero. The effect also diminishes with time and source size. Our results thus do not support claims in the literature that local dispersion must always be accounted for, no matter how small it is. First-order decay reduces dispersion. This effect increases with time. However, these concentration moments (c)(omega) and (c＇c＇)(omega) of total concentration c, which are associated with the scale below omega, cannot be used to estimate the field-scale concentration c(omega) directly. To do so, a spatial average over the field measurement scale omega is needed. Nevertheless, our numerical results show that differences between the ensemble moments of c(omega), and those of c are negligible, especially for nonpoint sources, because the ensemble moments of c are already smooth enough.