Accelerator magnets that can reach magnetic fields well beyond the Nb-Ti performance limits are presently being built and developed, using Nb$_3$Sn superconductors. This technology requires reaction heat treatment (RHT) of the magnet coils, during which Nb$_3$Sn is formed from its ductile precursor materials (a “wind and react” approach). The Nb$_3$Sn microstructure and microchemistry are strongly influenced by the conductor fabrication route, and by the phase changes during RHT. By combining in situ differential scanning calorimetry, high energy synchrotron x-ray diffraction, and micro-tomography experiments, we have acquired a unique data set that describes in great detail the phase and microstructure changes that take place during the processing of restacked rod process (RRP), powder-in-tube (PIT), and internal tin (IT) Nb$_3$Sn wires. At temperatures below 450 °C the phase evolutions in the three wire types are similar, with respectively solid state interdiffusion of Cu and Sn, Cu$_6$Sn$_5$ formation, and Cu$_6$Sn$_5$ peritectic transformation. Distinct differences in phase evolutions in the wires are found when temperatures exceed 450 °C. The volume changes of the conductor during RHT are a difficulty in the production of Nb$_3$Sn accelerator magnets. We compare the wire diameter changes measured in situ by dilatometry with the phase and void volume evolution of the three types of Nb$_3$Sn wire. Unlike the Nb$_3$Sn wire length changes, the wire diameter evolution is characteristic for each Nb$_3$Sn wire type. The strongest volume increase, of about 5%, is observed in the RRP wire, where the main diameter increase occurs above 600 °C upon Nb$_3$Sn formation.