The overall opinion is that this paper provides a valuable theoretical contribution, and I recommend accept. However, some reviewers (R4) have expressed concerns about the practical and implementation aspect of the work, and it is important that the authors implement the updates that they have promised in the rebuttal for the final version. Moreover, the lack of clarity raised by reviewer 1 (Q2) was not satisfyingly addressed (see his/her updated review: you are actually computing the same gradient flow (up to the kernelized distance), but the advantage of your method is that it is computationally cheaper than WFR). It is important to take those remarks into account in the final version to write a rigorous discussion.