Grounded dialogue models generate responses that are grounded on certain concepts. Limited by the distribution of grounded dialogue data, models trained on such data face the transferability challenges in terms of the data distribution and the type of grounded concepts. To address the challenges, we propose the grounded minimal editing framework, which minimally edits existing responses to be grounded on the given concept. Focusing on personas, we propose Grounded Minimal Editor (GME), which learns to edit by disentangling and recombining persona-related and persona-agnostic parts of the response. To evaluate persona-grounded minimal editing, we present the PersonaMi-nEdit dataset, and experimental results show that GME outperforms competitive baselines by a large margin. To evaluate the transferability, we experiment on the test set of BlendedSkillTalk and show that GME can edit dialogue models’ responses to largely improve their persona consistency while preserving the use of knowledge and empathy.