Remediation schemes for contaminated sites are often evaluated to assess their potential for source zone reduction of mass, or treatment of the contaminant between the source and a control plane (CP) to achieve regulatory limits. In this study, we utilize a stochastic stream tube model to explain the behavior of breakthrough curves (BTCs) across a CP. At the local scale, mass dissolution at the source is combined with an advection model with first-order decay for the dissolved plume. Field-scale averaging is then employed to account for spatial variation in mass within the source zone, and variation in the velocity field. Under the assumption of instantaneous mass transfer from the source to the moving liquid, semi-analytical expressions for the BTC and temporal moments are developed, followed by derivation of expressions for effective velocity, dispersion, and degradation coefficients using the method of moments. It is found that degradation strongly influences the behavior of moments and the effective parameters. While increased heterogeneity in the velocity field results in increased dispersion, degradation causes the center of mass of the plume to shift to earlier times, and reduces the dispersion of the BTC by lowering the concentrations in the tail. Modified definitions of effective parameters are presented for degrading solutes to account for the normalization constant (zeroth moment) that keeps changing with time or distance to the CP. It is shown that anomalous dispersion can result for high degradation rates combined with wide variation in velocity fluctuations. Implications of model results on estimating cleanup times and fulfillment of regulatory limits are discussed. Relating mass removal at the source to flux reductions past a control plane is confounded by many factors. Increased heterogeneity in velocity fields causes mass fluxes past a control plane to persist, however, aggressive remediation between the source and CP can reduce these fluxes.