The potential of AlxIn1-xAs/InP as a flexible candidate for future device applications relies on a thorough understanding of the growth mechanisms of this system during metalorganic vapor-phase epitaxy (MOVPE). We present a systematic study of the morphology of InP films grown on macroscopically lattice-matched AlxIn1-xAs during low-pressure MOVPE. InP and AlInAs alloys are often assumed to be fully compatible, as long as macroscopic lattice matching is preserved. However, thin InP films show a variety of spontaneously formed nanostructures as the growth conditions are varied. We have examined the systematic variation of individual growth conditions (temperature, V/III ratios, growth rate, and misalignment angle and direction) to obtain a comprehensive picture of the morphological evolution of this system. The analysis of these experiments is based on a kinetic model of the deposition and decomposition of polyatomic precursors and the subsequent adatom kinetics, which is used to form characteristic lengths from combinations of kinetic coefficients associated with each process. The orderings of these lengths for particular ranges of growth conditions account for many aspects of the morphological evolution on perfectly oriented and misoriented surfaces. For perfectly oriented surfaces, differences in surface free energies are suggested as the driving force for Volmer-Weber growth and the observed formation of multilayer features. Finally, we examine the transition from quantum rings to quantum dots during postgrowth recovery when the group-V source is switched from phosphine to arsine. A qualitative explanation of similar observations in InAs/InP based on As/P exchange is consistent with our findings.