The paper demonstrates the phenomenology of shock-driven air bubble and triple-layer microbubble collapse near tissue-like gelatin by direct numerical simulation. We validate the numerical method by comparison with experimental data for water jet generated by shock-driven bubble collapse near a gelatin interface. Coupled dynamics of bubble collapse and gelatin deformation is revealed during the early stages of the process. As the shock wave overpressure decreases, the coupling effect becomes more pronounced. The phenomenology of agent delivery initiated by shock-driven triple-layer microbubble collapse is presented. The results show that increasing the shock overpressure or reducing the agent layer thickness significantly enhances delivery efficiency, with the attached triple-layer microbubble exhibiting the highest efficiency. The results show how process parameters settings in shock-driven multi-material microcapsule collapse can control agent delivery processes in biomedical therapies.