Model merging (e.g., via interpolation or task arithmetic) fuses multiple models trained on different tasks to generate a multi-task solution. The technique has been proven successful in previous studies, where the models are trained on similar tasks and with the same initialization. In this paper, we expand on this concept to a multimodal setup by merging transformers trained on different modalities. Furthermore, we conduct our study for a novel goal where we can merge vision, language, and cross-modal transformers of a modality-specific architecture to create a parameter-efficient modality-agnostic architecture. Through comprehensive experiments, we systematically investigate the key factors impacting model performance after merging, including initialization, merging mechanisms, and model architectures. We also propose two metrics that assess the distance between weights to be merged and can serve as an indicator of the merging outcomes. Our analysis leads to an effective training recipe for matching the performance of the modality-agnostic baseline (i.e., pre-trained from scratch) via model merging. Our method also outperforms naive merging significantly on various tasks, with improvements of 3% on VQA, 7% on COCO retrieval, 25% on NLVR2, 14% on Flickr30k and 3% on ADE20k.