Real-time interaction with multimodal Large Language Models (M-LLMs) remains impractical because conventional serving pipelines impose a receive-then-process barrier: video must be fully uploaded and preprocessed before inference can begin, leading to accelerator cold-start and high end-to-end latency. We present RabbitLLM, an end-to-end serving architecture that enables network-compute pipelining by coupling progressive video transport with transport-triggered incremental KV-cache construction. RabbitLLM streams video frames using WebRTC and immediately prefixes arriving visual chunks into the model state, overlapping transmission with prefilling while deferring decoding until a warm cache is available. Across realistic 4G/5G traces, RabbitLLM reduces interactive end-to-end delay by 43.5% compared to an upload-first TCP baseline while preserving output quality (BERTScore F1 > 0.82). RabbitLLM also reduces per-frame transmission delay by 95.4% and bandwidth consumption by 99.1%. These results suggest that low-latency multimodal serving requires co-design between streaming transport and inference execution, rather than optimizing either layer in isolation.