Measurements of radiogenic isotopes can in principle constrain the melting, melt migration, and solid state convection that occurs in the Earth's mantle, but to do so requires suitable quantitative models. A new statistical model is introduced to better understand the observed heterogeneity in isotopic ratios 143Nd/144Nd, 87Sr/86Sr, 176Hf/177Hf,208Pb/204Pb, 206Pb/204Pb and 207Pb/204Pb measured on mid-ocean ridge basalt. The model is highly idealised, analytically tractable, and contains the essential physical processes involved: radioactive decay, the stirring and recycling of mantle convection, partial melting, and the mixing of melts. Comparison of the modelled heterogeneity with that observed constrains model parameters, which in turn constrains aspects of mantle convection and melting. The model provides a new interpretation of the 2.0 Ga lead-lead pseudo-isochron age in terms of an age distribution of mantle material. Simple equations relate the pseudo-isochron age to the rate of melting and decay constants. These equations are different from, but related to and more general than, those found previously for standard geochemical box models. The results are in good agreement with numerical simulations of mantle convection. The 2.0 Ga pseudo-isochron age is shown to infer a 0.5 Ga average time scale for melting of mantle material. Geochemical and geological evidence suggests that melt travels to the surface via a network of channels under the ridge. Motivated by this, the fluid dynamical problem of a open melt conduit surrounded by a deformable porous medium is studied. Previous work has shown that the conduit supports solitary waves of elevation, with a region of trapped melt travelling with the wave. The new analysis comes to a different conclusion, showing that the solitary wave is instead one of depression, without a region of trapped melt.