This article proposes goodness-of-fit tests based on a linear transformation of Barlett’s Tp-process, which converges weakly to the standard Brownian motion under the null hypothesis, in the lines suggested by Khmaladze (1981). We show that the proposed test has non trivial power in the direction of contiguous alternatives converging to the null at the parametric rate n-1/2. A Monte Carlo study illustrates the performance of the test with moderate sample sizes, showing that it works fairly well, in terms of size accuracy and power. We also show that this testing procedure can be applied to test the specification of general nonstationary integrated processes, as well as lack-of-autocorrelation of the innovations in general dynamic models.