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One hypothesis for the origin of alkaline lavas erupted on oceanic islands and in intracontinental settings is that they represent the melts of amphibole-rich veins in the lithosphere (or melts of their dehydrated equivalents if metasomatized lithosphere is recycled into the convecting mantle). Amphibole-rich veins are interpreted as cumulates produced by crystallization of low-degree melts of the underlying asthenosphere as they ascend through the lithosphere. We present the results of trace-element modelling of the formation and melting of veins formed in this way with the goal of testing this hypothesis and for predicting how variability in the formation and subsequent melting of such cumulates (and adjacent cryptically and modally metasomatized lithospheric peridotite) would be manifested in magmas generated by such a process. Because the high-pressure phase equilibria of hydrous near-solidus melts of garnet lherzolite are poorly constrained and given the likely high variability of the hypothesized accumulation and remelting processes, we used Monte Carlo techniques to estimate how uncertainties in the model parameters (e.g. the compositions of the asthenospheric sources, their trace-element contents, and their degree of melting; the modal proportions of crystallizing phases, including accessory phases, as the asthenospheric partial melts ascend and crystallize in the lithosphere; the amount of metasomatism of the peridotitic country rock; the degree of melting of the cumulates and the amount of melt derived from the metasomatized country rock) propagate through the process and manifest themselves as variability in the trace-element contents and radiogenic isotopic ratios of model vein compositions and erupted alkaline magma compositions. We then compare the results of the models with amphibole observed in lithospheric veins and with oceanic and continental alkaline magmas. While the trace-element patterns of the near-solidus peridotite melts, the initial anhydrous cumulate assemblage (clinopyroxene ± garnet ± olivine ± orthopyroxene), and the modelled coexisting liquids do not match the patterns observed in alkaline lavas, our calculations show that with further crystallization and the appearance of amphibole (and accessory minerals such as rutile, ilmenite, apatite, etc.) the calculated cumulate assemblages have trace-element patterns that closely match those observed in the veins and lavas. These calculated hydrous cumulate assemblages are highly enriched in incompatible trace elements and share many similarities with the trace-element patterns of alkaline basalts observed in oceanic or continental setting such as positive Nb/La, negative Ce/Pb, and similiar slopes of the rare earth elements. By varying the proportions of trapped liquid and thus simulating the cryptic and modal metasomatism observed in peridotite that surrounds these veins, we can model the variations in Ba/Nb, Ce/Pb, and Nb/U ratios that are observed in alkaline basalts. If the isotopic compositions of the initial low-degree peridotite melts are similar to the range observed in mid-ocean ridge basalt, our model calculations produce cumulates that would have isotopic compositions similar to those observed in most alkaline ocean island basalt (OIB) and continental magmas after ~0·15 Gyr. However, to produce alkaline basalts with HIMU isotopic compositions requires much longer residence times (i.e. 1–2 Gyr), consistent with subduction and recycling of metasomatized lithosphere through the mantle. EM magmas cannot readily be explained without appealing to other factors such as a heterogeneous asthenosphere. These modelling results support the interpretation proposed by various researchers that amphibole-bearing veins represent cumulates formed during the differentiation of a volatile-bearing low-degree peridotite melt and that these cumulates are significant components of the sources of alkaline OIB and continental magmas. The results of the forward models provide the potential for detailed tests of this class of hypotheses for the origin of alkaline magmas worldwide and for interpreting major and minor aspects of the geochemical variability of these magmas.