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 suboptimal 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, realizing four distinct morphologies of the as-grown compounds. We use this to determine optimized growth conditions for the fabrication of high-quality monolayers, ultimately realizing the largest grain sizes of monolayer TMDs that have been achieved to date via MBE growth.