Pharmaceuticals and heavy metals pose significant risks to human health and aquatic ecosystems, necessitating their removal from water and wastewater. A promising alternative for this purpose involves their removal by adsorption on composite sorbents prepared using a conventional layer-by-layer (LbL) method or an innovative coacervate direct deposition approach. In this study, four novel composite materials based on a silica core (IS) and a polyelectrolyte coacervate shell were used for the investigation of dynamic adsorption of three heavy metals (lead, nickel and cadmium) and an organic drug model (diclofenac sodium salt, DCF-Na). The four types of composite sorbents were tested for the first time in dynamic conditions (columns with continuous flow), and the column conditions were similar to those used in wastewater treatment plants. The influence of the polyanion nature (poly(acrylic acid) (PAA) vs. poly(sodium methacrylate) (PMAA)), maintaining a constant poly(ethyleneimine) (PEI), and the cross-linking degree (r = 0.1 and r = 1.0) of PEI chains on the immobilization of these pollutants (inorganic vs. organic) on the same type of composite was also studied. The experiments involved both single- and multi-component aqueous solutions. The kinetics of the dynamic adsorption process were examined using two non-linear models: the Thomas and Yoon–Nelson models. The tested sorbents demonstrated good adsorption capacities with affinities for the metal ions in the following order: Pb2+ > Cd2+ > Ni2+. An increase in the initial diclofenac sodium concentration led to an enhanced adsorption capacity of the IS/(PEI-PAA)c-r1 sorbent. The calculated sorption capacities were in good agreement with the adsorption capacity predicted by the Thomas and Yoon–Nelson models. The substantial affinity observed between DCF-Na and a column containing composite microparticles saturated with heavy metal ions was explained.