For sure there is a natural reparameterization of the $\gamma_\epsilon$ but to be honest I do not see how this can be related to the Borel regularity of the reparametrization operator. In a sense, if you want, we can directly consider the curve in $\mathbb R^{d+1}$ defined by $(h, \gamma)$. But how can this be used to prove the Borel regularity of $\mathcal R$?

@Rbega Thanks for your comments! Yes, exactly, that was a typo, I fixed it. Not precise applications in mind, it is a question which arose several times and I believe there must be somewhere in the literature a general result of this kind. I am not sure of getting your hint: what is the point in considering the curve you suggest (and using arc-length there)? Thanks.

@PietroMajer Thanks again for your kind reply! Exactly, that was also an idea I considered but I got stuck because: 1. I am not completely sure of having an argument to prove that (inverses) of strictly monotone reparameterizations induce Borel maps between curves: do you have any references for this? 2. I had not a clear idea of how the perturbations converge (pointwise?); 3. I thought there has to be a general (well-known) argument behind (thus I came here to ask). Thanks for your valuable comments.

Thus I am sorry, but I cannot accept your answer.

Thanks for the useful comment. Yes, I have been imprecise. Let me add that I am assuming the association $\gamma \mapsto h_{\gamma}$ is Borel (between the space of Lipschitz curves in $\mathbb R^d$ and the Lipschitz maps in $\mathbb R$). Say now that I define $s_\gamma$ to be $s_\gamma(r) = \inf \{t: h_\gamma(t)>r \}$ (this should be the left inverse of $h_\gamma$). Is the corresponding association $\gamma \mapsto s_\gamma$ now at least Borel? Thanks again.

Thanks a lot for your interest and for your references. I admit I have not a solid background in differential geometry, thus I am a bit lost in the huge book you mention. I went through the third chapter and some of the others but I did not find anything specifically related to my problem. The same for the paper you mention and another paper by Michor I have seen: it seems they do not discuss the regularity (at the level of measurability at least, which is the one I need) of the reparameterization maps. Could you be so kind to indicate me a more precise reference therein, please? Thanks!