Mobile video analysis systems often encounter various deploying environments, where environment shifts present greater demands for responsiveness in adaptations of deployed "expert DNN models". Existing model adaptation frameworks primarily operate in a cloud-centric way, exhibiting degraded performance during adaptation and delayed reactions to environment shifts. Instead, this paper proposes MOCHA, a novel framework optimizing the responsiveness of continuous model adaptation through hierarchical collaborations between mobile and cloud resources. Specifically, MOCHA (1) reduces adaptation response delays by performing on-device model reuse and fast fine-tuning before requesting cloud model retrieval and end-to-end retraining; (2) accelerates history expert model retrieval by organizing them into a structured taxonomy utilizing domain semantics analyzed by a cloud foundation model as indices; (3) enables efficient local model reuse by maintaining onboard expert model caches for frequent scenes, which proactively prefetch model weights from the cloud model database. Extensive evaluations with real-world videos on three DNN tasks show MOCHA improves the model accuracy during adaptation by up to 6.8% while saving the response delay and retraining time by up to 35.5x and 3.0x respectively.