To advance fundamental understanding, and ultimate application, of transition-metal dichalcogenide (TMD) monolayers, it is essential to develop capabilities for the synthesis of high-quality single-layer samples. Molecular beam epitaxy (MBE), a leading technique for the fabrication of the highest-quality epitaxial films of conventional semiconductors has, however, typically yielded only small grain sizes and sub-optimal morphologies when applied to the van der Waals growth of monolayer TMDs. Here, we present a systematic study on the influence of adatom mobility, growth rate, and metal:chalcogen flux on the growth of NbSe₂, VSe₂ and TiSe₂ using MBE. Through this, we identify the key drivers and influence of the adatom kinetics that control the epitaxial growth of TMDs, realising four distinct morphologies of the as-grown compounds. We use this to determine optimised growth conditions for the fabrication of high-quality monolayers, ultimately realising the largest grain sizes of monolayer TMDs that have been achieved to date via MBE growth.