Conversational search aims to satisfy users’ complex information needs via multiple-turn interactions. The key challenge lies in revealing real users’ search intent from the context-dependent queries. Previous studies achieve conversational search by fine-tuning a conversational dense retriever with relevance judgments between pairs of context-dependent queries and documents. However, this training paradigm encounters data scarcity issues. To this end, we propose ConvMix, a mixed-criteria framework to augment conversational dense retrieval, which covers more aspects than existing data augmentation frameworks. We design a two-sided relevance judgment augmentation schema in a scalable manner via the aid of large language models. Besides, we integrate the framework with quality control mechanisms to obtain semantically diverse samples and near-distribution supervisions to combine various annotated data. Experimental results on five widely used benchmarks show that the conversational dense retriever trained by our ConvMix framework outperforms previous baseline methods, which demonstrates our superior effectiveness.