Site-controlled quantum dots formed during the deposition of (Al)GaAs layers by metal-organic vapor-phase epitaxy on GaAs(111)B substrates patterned with inverted pyramids result in geometric and compositional self-ordering along the vertical axis of the template. We describe a theoretical scheme that reproduces the experimentally observed time-dependent behavior of this process, including the evolution of the recess and the increase of Ga incorporation along the base of the template to stationary values determined by alloy composition and other growth parameters. Our work clarifies the interplay between kinetics and geometry for the development of self-ordered nanostructures on patterned surfaces, which is essential for the reliable on-demand design of confined systems for applications to quantum optics.