Large-scale pre-trained text-image models with dual-encoder architectures (such as CLIP) are typically adopted for various vision-language applications, including text-image retrieval. However,these models are still less practical on edge devices or for real-time situations, due to the substantial indexing and inference time and the large consumption of computational resources. Although knowledge distillation techniques have been widely utilized for uni-modal model compression, how to expand them to the situation when the numbers of modalities and teachers/students are doubled has been rarely studied. In this paper, we conduct comprehensive experiments on this topic and propose the fully-Connected knowledge interaction graph (Cona) technique for cross-modal pre-training distillation. Based on our findings, the resulting ConaCLIP achieves SOTA performances on the widely-used Flickr30K and MSCOCO benchmarks under the lightweight setting. An industry application of our method on an e-commercial platform further demonstrates the significant effectiveness of ConaCLIP.