Clinical machine learning deployment across institutions faces significant challenges when patient populations and clinical practices differ substantially. We present a systematic framework for cross-institutional knowledge transfer in clinical time series, demonstrated through pediatric ventilation management between a general pediatric intensive care unit (PICU) and a cardiac-focused unit. Using contrastive predictive coding (CPC) for representation learning, we investigate how different data regimes and fine-tuning strategies affect knowledge transfer across institutional boundaries. Our results show that while direct model transfer performs poorly, CPC with appropriate fine-tuning enables effective knowledge sharing between institutions, with benefits particularly evident in limited data scenarios. Analysis of transfer patterns reveals an important asymmetry: temporal progression patterns transfer more readily than point-of-care decisions, suggesting practical pathways for cross-institutional deployment. Through a systematic evaluation of fine-tuning approaches and transfer patterns, our work provides insights for developing more generalizable clinical decision support systems while enabling smaller specialized units to leverage knowledge from larger centers.