Using the same approach introduced in a previous paper to describe the transition between incomplete fusion and multifragmentation processes, we propose to extend our schematic phenomenological model to non-zero impact-parameter collisions. The approach is based on a two-step process. In the first step, the thermalization and the emission of prompt particles are described using a preequilibrium model. In the second step, the evolution of the thermalized system is followed with a simple isentropic model where the magnitude of the fluctuations of the mean field are checked at each stage of the evolution using a site-bond percolation model on a 3-dimensional lattice. Multifragmentation corresponds to the case where these fluctuations become large. In this case, the system breaks up in several pieces. Otherwise, the projectile and the target may fuse. In the present work, a dynamical model and geometric considerations allow us to study the influence of the impact parameter of the collision. Moreover, since we are dealing with systems containing a small number of particles, we show that the transition between incomplete fusion and multifragmentation is smooth and it extends over a broad incident energy range. Finally, we calculate both incomplete fusion and multifragmentation excitation functions for collisions induced by C, Ne, Ar beams on Ag and Au targets. Together with fragment multiplicities, the obtained results can be directly compared with experimental data. For the sake of completeness, we have derived analytical expressions for most of the quantities which are calculated in the model. In the appendix, we describe a general procedure to obtain a complete set of results.