This study develops an analytical description of the motion of dilute solid particles in the boundary layer of laminar horizontal flows subjected to weak transverse pulsations. A coupled matrix representation of the governing equations is formulated, and closed-form solutions are obtained using Laplace transformation. The analytical expressions capture transient evolution, forced oscillations, resonance effects, and long-term behaviour for particles with different density ratios. Numerical evaluation shows that light particles migrate toward faster regions of the boundary layer and accelerate longitudinally, while heavy particles move toward slower layers and decelerate. Transverse pulsations generate oscillatory trajectories whose amplitude increases near resonance. Impulsive perturbations superimposed on the continuous motion lead to discontinuous transitions consistent with the linear matrix system. The results provide a unified physical interpretation of particle redistribution mechanisms in boundary layers and offer a compact analytical tool for dilute multiphase flow modelling.