The chemistry of magmas erupted by volcanoes is a message from deep within the Earth's crust, which if decrypted, can provide essential information on magmatic processes occurring at inaccessible depths. While some volcanoes are prone to erupt magmas of a wide compositional variety, others sample rather monotonous chemistries through time. Whether such differences are a consequence of physical filtering or reflect intrinsic properties of different magmatic systems remains unclear. Here we show, using thermal and petrological modelling, that magma flux and the thermal structure of the crust modulate diversity and temporal evolution of magma chemistry in mid to deep crustal reservoirs. Our analysis shows that constant rates of magma input leads to extractable magma compositions that tend to evolve from felsic to more mafic in time. Low magma injection rates into hot or deep crust produces less chemical variability of extractable magma compared to the injection of large batches in colder or shallower crust. Our calculations predict a correlation between magma fluxes and compositional diversity that resembles trends observed in volcanic deposits. Our approach allows retrieval of quantitative information about magma input and the thermal architecture of magmatic systems from the chemical diversity and temporal evolution of volcanic products.