Large deep neural networks (DNNs), especially transformer-based and multimodal architectures, are computationally demanding and challenging to deploy on resource-constrained edge platforms like field robots. These challenges intensify in mission-critical scenarios (e.g., disaster response), where robots must collaborate under tight constraints on bandwidth, latency, and battery life, often without infrastructure or server support. To address these limitations, we present COHORT, a collaborative DNN inference and task-execution framework for multi-robot systems built on the Robotic Operating System (ROS). COHORT employs a hybrid offline-online reinforcement learning (RL) strategy to dynamically schedule and distribute DNN module execution across robots. Our key contributions are threefold: (a) Offline RL policy learning combined with Advantage-Weighted Regression (AWR), trained on auction-based task allocation data from heterogeneous DNN workloads across distributed robots, (b) Online policy adaptation via Multi-Agent PPO (MAPPO), initialized from the offline policy and fine-tuned in real time, and (c) comprehensive evaluation of COHORT on vision-language model (VLM) inference tasks such as CLIP and SAM, analyzing scalability with increasing robot/workload and robustness under . We benchmark COHORT against genetic algorithms and multiple RL baselines. Experimental results demonstrate that COHORT reduces battery consumption by 15.4% and increases GPU utilization by 51.67%, while satisfying frame-rate and deadline constraints 2.55 times of the time.