  We explicitly construct parameter transformations between gradient flows in
metric spaces, called curves of maximal slope, having different exponents when
the associated function satisfies a suitable convexity condition. These
transformations induce the uniqueness of gradient flows for all exponents under
a natural assumption which is satisfied in many examples. We also prove the
regularizing effects of gradient flows. To establish these results, we directly
deal with gradient flows instead of using variational discrete approximations
which are often used in the study of gradient flows.
