We present an integral-equation solution of the structure of systems built through the sequential quenching of particles. The theory is based on the Replica Ornstein-Zernike equations that describe the structure of equilibrium fluids within random porous matrices. The quenched particles are treated as a polydisperse system, each of them labeled by the total density at the time of its arrival. The diagrammatic expansions of the correlation functions lead to the development of the liquid-theory closures appropriate for the present case. Numerical solutions for the deposition of hard disks show excellent agreement with simulation. We also discuss a binary-mixture treatment, which is shown to provide a very good approximation to the polydisperse approach. (C) 2000 American Institute of Physics. [S0021-9606(00)50314-6].