Most existing methods for testing equality of means of functional data from multiple populations rely on assumptions of equal covariance and/or Gaussianity. In this work we provide a new testing method based on a statistic that is distribution-free under the null hypothesis (i.e. the statistic is pivotal), and allows different covariance structures across populations, while Gaussianity is not required. In contrast to classical methods of functional mean testing, where either observations of the full curves or projections are applied, our method allows the projection dimension to increase with the sample size to allow asymptotic recovery of full information as the sample size increases. We obtain a unified theory for the asymptotic distribution of the test statistic under local alternatives, in both the sample and bootstrap cases. The finite sample performance for both size and power have been studied via simulations and the approach has also been applied to two real datasets.