Large vision-language models (LVLMs) have achieved impressive performance across multimodal tasks, but their reliance on visual inputs exposes them to adversarial threats. Encoder-based attacks provide an efficient alternative to end-to-end optimization by crafting perturbations through the vision encoder alone. However, existing encoder-based attacks often assume that the surrogate encoder is identical or similar to the victim LVLM's vision encoder. In this work, we present a systematic study of their transferability in more realistic black-box deployments with heterogeneous LVLM architectures. We find that model-specific visual evidence is inconsistent across models, whereas text-conditioned grounding regions are more closely tied to caption-relevant evidence and provide a more stable transfer target. However, existing attacks remain weakly aligned with and insufficiently disrupt these regions. Motivated by these findings, we propose Grounding-Driven Attack (GDA), which aligns perturbation optimization with text-grounded evidence. GDA combines Grounding-Aware Perturbation Allocation to concentrate perturbation budget on grounded evidence regions with Grounding-Centric Evidence Disruption to intensify their global and local disruption. Experiments across diverse victim models and tasks show that GDA consistently outperforms existing encoder-based attacks in black-box transfer. These results highlight the central role of text-grounded evidence in adversarial transferability and motivate grounding-aware robustness evaluation and defense design.