Fast domain adaptation remains a fundamental challenge for deploying multi-agent systems across diverse environments in Vehicle-to-Everything (V2X) collaborative perception. Despite the success of Parameter-Efficient Fine-Tuning (PEFT) in natural language processing and conventional vision tasks, directly applying PEFT to multi-agent settings leads to significant performance degradation and training instability. In this work, we conduct a detailed analysis and identify two key factors: (i) inter-frame redundancy in heterogeneous sensory streams, and (ii) erosion of fine-grained semantics in deep-layer representations under PEFT adaptation. To address these issues, we propose FlowAdapt, a parameter-efficient framework grounded in optimal transport theory, which minimizes information transport costs across both data distributions and network hierarchies. Specifically, we introduce a Wasserstein Greedy Sampling strategy to selectively filter redundant samples via a bounded covering radius. Furthermore, Progressive Knowledge Transfer module is designed to progressively inject compressed early-stage representations into later stages through learnable pathways, alleviating semantic degradation in late-stage adaptation. Extensive experiments on three benchmarks demonstrate that FlowAdapt achieves state-of-the-art performance with only 1% of trainable parameters, effectively bridging domain gaps with superior sample efficiency and generalization.