The"Breathing Lung-on-a-Chip,"a novel microfluidic device featuring a stretchable membrane, replicates the natural expansion and contraction of the human lung. It provides a more realistic in-vitro platform to study respiratory diseases, particle deposition, and drug delivery mechanisms. This device enables investigations into the effects of inhaled nanoparticles (NPs) on lung tissue and supports the development of advanced inhalation therapies. Uniform and optimal concentration delivery of NPs to cultured cells within the chip is critical, particularly as membrane stretching significantly influences particle dynamics. To address this, we developed a 3D numerical model that accurately simulates NP behavior under dynamic conditions, overcoming experimental limitations. The model, validated against experimental data, explores the effects of flow dynamics, particle size, membrane porosity, and stretching frequency/intensity on NP deposition in the air channel and transfer through the porous membrane into the medium channel. The results indicate that increased membrane stretch enhances the sedimentation rate of NPs in the air channel, thereby promoting their transfer to the medium channel, particularly in membranes with initially low porosity. Additionally, excessive stretching frequencies or intensities can introduce reverse flow and stagnation, leading to a longer residence time for NPs and altering their sedimentation patterns. These insights advance our understanding of NP transport in dynamic lung environments, paving the way for more effective applications of lung-on-a-chip technology in toxicological assessments and respiratory therapy innovations.