As a general-purpose vision-language pretraining model, CLIP demonstrates strong generalization ability in image-text alignment tasks and has been widely adopted in downstream applications such as image classification and image-text retrieval. However, it struggles with fine-grained tasks such as object detection and semantic segmentation. While many variants aim to improve CLIP on these tasks, its robustness to adversarial perturbations remains underexplored. Understanding how adversarial examples transfer across tasks is key to assessing CLIP's generalization limits and security risks. In this work, we conduct a systematic empirical analysis of the cross-task transfer behavior of CLIP-based models on image-text retrieval, object detection, and semantic segmentation under adversarial perturbations. We find that adversarial examples generated from fine-grained tasks (e.g., object detection and semantic segmentation) often exhibit stronger transfer potential than those from coarse-grained tasks, enabling more effective attacks against the original CLIP model. Motivated by this observation, we propose a novel framework, Multi-Task Adversarial CLIP (MT-AdvCLIP), which introduces a task-aware feature aggregation loss and generates perturbations with enhanced cross-task generalization capability. This design strengthens the attack effectiveness of fine-grained task models on the shared CLIP backbone. Experimental results on multiple public datasets show that MT-AdvCLIP significantly improves the adversarial transfer success rate (The average attack success rate across multiple tasks is improved by over 39%.) against various CLIP-derived models, without increasing the perturbation budget. This study reveals the transfer mechanism of adversarial examples in multi-task CLIP models, offering new insights into multi-task robustness evaluation and adversarial example design.