LLM-based multi-agent systems (LLM-MAS) have become a promising paradigm for solving complex tasks through role specialization, tool use, memory, and collaborative reasoning. However, these interactions create new security risks that malicious instructions injected through messages, tools, or memories can propagate across agents and rounds, causing system-level compromise. Existing defenses largely rely on local filtering or graph-based anomaly detection, but they often fail to trace fine-grained propagation paths or remediate contaminated states without disrupting benign collaboration. We propose PropGuard, a propagation-aware framework for safeguarding LLM-MAS. PropGuard constructs a dual-view spatio-temporal graph that combines response-centric risk estimation with full-state evidence preservation. Guided by these risk priors, a GE-GRPO trained inspector sequentially explores the full-state graph to recover compact suspicious propagation subgraphs. PropGuard then verifies harmful propagation through subgraph-aware diagnosis and applies source-guided remediation to correct upstream contamination and replay affected downstream interactions. Experiments across four communication architectures and five attack settings demonstrate that PropGuard consistently lowers attack success while maintaining high task-level defense success, achieving a favorable effectiveness--efficiency trade-off.