We explore the statistics of assembling soft-matter building blocks to investigate the uptake and encapsulation of cargo particles by carriers engulfing their load. While the such carrier-cargo complexes are important for many applications out of equilibrium, such as drug delivery and synthetic cell encapsulation, we uncover here the basic statistical physics in minimal hard-core-like models for particle uptake. Introducing an exactly solvable equilibrium model in one dimension, we demonstrate that the formation of carrier-cargo complexes can be largely tuned by both the cargo concentration and the carriers' interior size. These findings are intuitively explained by interpreting the internal free space (partition function) of the cargo inside a carrier as its engulfment strength, which can be mapped to an external control parameter (chemical potential) of an additional effective particle species. Assuming a hard carrier membrane, such a mapping can be exactly applied to account for multiple cargo uptake involving various carrier or cargo species and even attractive uptake mechanisms, while soft interactions require certain approximations. We further argue that the Boltzmann occupation law identified within our approach is broken when particle uptake is governed by nonequilibrium forces. Speculating on alternative occupation laws using effective parameters, we put forward a Bose-Einstein-like phase transition associated with polydisperse carrier properties.