Skill-based prompting has become a practical mechanism for improving LLM agents, yet existing methods often treat skills as summaries of the agent's own experience or of successful demonstrations. This creates a mismatch for weaker student agents. A failed trajectory may not reveal the missing knowledge or strategy, while a teacher trajectory may be too implicit to internalize. We propose SKILL-KD, a contrastive skill distillation framework that treats skills as an explicit distillation medium between agents of different capabilities. Given a student failure and the teacher trajectory on the same task, SKILL-KD distills their actionable discrepancy into a textual skill patch, evaluates it by re-running the student, and iteratively refines it when the student still fails. To prevent skill drift from repeated local updates, SKILL-KD maintains trace-linked edit histories and performs Drift-Aware Skill Consolidation to decide whether each patch is added, merged, or skipped. Across five agent benchmarks and two student settings, SKILL-KD consistently improves frozen student agents over fixed-model adaptation baselines.

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