The infant brain undergoes rapid development in the first few years after birth.Compared to cross-sectional studies, longitudinal studies can depict the trajectories of infants brain development with higher accuracy, statistical power and flexibility.However, the collection of infant longitudinal magnetic resonance (MR) data suffers a notorious dropout problem, resulting in incomplete datasets with missing time points. This limitation significantly impedes subsequent neuroscience and clinical modeling. Yet, existing deep generative models are facing difficulties in missing brain image completion, due to sparse data and the nonlinear, dramatic contrast/geometric variations in the developing brain. We propose LoCI-DiffCom, a novel Longitudinal Consistency-Informed Diffusion model for infant brain image Completion,which integrates the images from preceding and subsequent time points to guide a diffusion model for generating high-fidelity missing data. Our designed LoCI module can work on highly sparse sequences, relying solely on data from two temporal points. Despite wide separation and diversity between age time points, our approach can extract individualized developmental features while ensuring context-aware consistency. Our experiments on a large infant brain MR dataset demonstrate its effectiveness with consistent performance on missing infant brain MR completion even in big gap scenarios, aiding in better delineation of early developmental trajectories.