Yeah, I meant with other people!

Let's say, the references in the other paper are somehow "scattered" among the introduction, while I put them all in the same place. I also checked each reference to be sure that it was really relevant

@mathworker21 I would say that "but you get tenure" is a non trivial statement, it very much depends on many factors (geographical for example)

@IosifPinelis I mean if using the Lipschitz constant on $f$ is the only known (at leat, well known) way to estimate the distance between the two push-forward measures, not requiring anything specific on $\mu$ and $\nu$

@IosifPinelis do you perhaps know some counterexample that shows that, if $f$ is not Lipschitz, we can say nothing on the distance between $f_\# \mu, f_\# \nu$? Or do you at least believe this to be true?

@Hannes unwissen thanks. Especially Ziemer's book goes in great detail, in a very general setting. Actually, I edited the question since I got carried away asking a question way more general than what I really need. Hope that in the simplified case presented there is a more explicit answer. It might be possible to recover it by going through Ziemer's book, but I am not 100% sure

Could you elaborate a little? In particular bounding the volume using the Hausdorff distance and the argument using the tube formula and the reach (never heard of neither of them to be honest)