Almost all previous text-to-video retrieval works ideally assume that videos are pre-trimmed with short durations containing solely text-related content. However, in practice, videos are typically untrimmed in long durations with much more complicated background content. Therefore, in this paper, we focus on the more practical yet challenging task of Partially Relevant Video Retrieval (PRVR), which aims to retrieve partially relevant untrimmed videos with the given query. To tackle this task, we propose a novel framework that distills generalization knowledge from a powerful large-scale vision-language pre-trained model and transfers it to a lightweight, task-specific PRVR network. Specifically, we introduce a Dual Learning framework with Dynamic Knowledge Distillation (DL-DKD++), where a large teacher model provides supervision to a compact dual-branch student network. The student model comprises two branches: an inheritance branch that absorbs transferable knowledge from the teacher, and an exploration branch that learns task-specific information from the PRVR dataset to address domain gaps. To further enhance learning, we incorporate a dynamic soft-target construction mechanism. By replacing rigid hard-target supervision with adaptive soft targets that evolve during training, our method enables the model to better capture the fine-grained, partial relevance between videos and queries. Experiment results demonstrate that our proposed model achieves state-of-the-art performance on TVR, ActivityNet, and Charades-STA datasets for PRVR. The code is available at https://github.com/HuiGuanLab/DL-DKD.