Accurate brain tumor segmentation is essential for preoperative evaluation and personalized treatment. Multi-modal MRI is widely used due to its ability to capture complementary tumor features across different sequences. However, in clinical practice, missing modalities are common, limiting the robustness and generalizability of existing deep learning methods that rely on complete inputs, especially under non-dominant modality combinations. To address this, we propose AdaMM, a multi-modal brain tumor segmentation framework tailored for missing-modality scenarios, centered on knowledge distillation and composed of three synergistic modules. The Graph-guided Adaptive Refinement Module explicitly models semantic associations between generalizable and modality-specific features, enhancing adaptability to modality absence. The Bi-Bottleneck Distillation Module transfers structural and textural knowledge from teacher to student models via global style matching and adversarial feature alignment. The Lesion-Presence-Guided Reliability Module predicts prior probabilities of lesion types through an auxiliary classification task, effectively suppressing false positives under incomplete inputs. Extensive experiments on the Pretreat-MetsToBrain-Masks and BraTS 2018, 2024 datasets demonstrate that AdaMM consistently outperforms existing methods, exhibiting superior segmentation accuracy and robustness, particularly in single-modality and weak-modality configurations. In addition, we conduct a systematic evaluation of six categories of missing-modality strategies, supporting the superiority of knowledge distillation and offering practical guidance for method selection and future research. Our source code is available at https://github.com/Quanato607/AdaMM.