The Khmaladze transform takes the vector of components of Pearson's chi-square statistic to another vector which contains the same "statistical information" but is asymptotically distribution-free. Hence any test statistic based on the new vector is also asymptotically distribution-free. Natural examples are goodness-of-fit statistics based on partial sums.
A version of the Khmaladze transform for testing whether data comes from a continuous distribution function, F, maps the normalized error function, which is asymptotically an F-bridge, to a process which is asymptotically a standard Brownian bridge. The associated statistic, which is empirically based, can be used for distribution-free testing.