Often the water flowing in a solution conduit is a combination of contaminated water entering at sinkholes and cleaner water released from the limestone matrix. The concentration of contaminants flowing in a conduit is reduced by this dilution and also by longitudinal conduit dispersion. This article seeks to quantify relative importance of these two mechanisms. Water entering the conduit from the matrix causes the conduit flow speed to increase with distance. This in turn causes the strength of dispersion to increase in the downstream direction. The breakthrough curve at a spring, resulting from transport, dilution, and dispersion, has been obtained for the initial-value problem using a modification of the standard Green's function approach, employing characteristic curves. The predicted breakthrough curves are skewed, with a rapid rise and slow decay. This feature does not result from the ordinary advection-dispersion model. Applying the new model to a dye-tracing experiment between Ames Sink and Indian Spring, Northwest Florida yields a value of dispersivity at 400 m. It is concluded that variable dispersion provides a possible explanation for the skewness and tailing that are often observed in spring breakthrough curves. It is demonstrated that the new model, in conjunction with the measured spring breakthrough curve, can extract the parameters of the conduit and flow.