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Large language models (LLMs) produce responses rated as highly empathic in single-turn settings (Ayers et al., 2023; Lee et al., 2024), yet they are also known to be formulaic generators that reuse the same lexical patterns, syntactic templates, and discourse structures across tasks (Jiang et al., 2025; Shaib et al., 2024; Namuduri et al., 2025). Less attention has been paid to whether this formulaicity extends to the level of discourse moves, i.e., what a response does for the person it is addressing. This question is especially consequential for empathic dialogue, where effective support demands not just a kind response at one moment but varied strategies as a conversation unfolds (Stiles et al., 1998). Indeed, prior work shows that LLMs reuse the same tactic sequences more than human supporters in single-turn settings (Gueorguieva et al., 2026). We extend this analysis to multi-turn conversations and find that the rigidity compounds: once a tactic appears in a supporter turn, LLMs reuse it in the next at nearly double the rate of humans (0.50-0.56 vs. 0.27). This pattern holds across LLMs serving as supporters in real emotional support conversations, and is invisible to standard similarity metrics. To address this gap, we introduce MINT (Multi-turn Inter-tactic Novelty Training), the first reinforcement learning framework to optimize discourse move diversity across multi-turn empathic dialogue. The best MINT variant combines an empathy quality reward with a cross-turn tactic novelty signal, improving aggregate empathy by 25.3% over vanilla across 1.7B and 4B models while reducing cross-turn discourse move repetition by 26.3% on the 4B model, surpassing all baselines including quality-only and token-level diversity methods on both measures. These results suggest that what current models lack is not empathy itself, but the ability to vary their discourse moves across a conversation.