Summary: The permutation approach for testing the equality of distributions and thereby comparing two populations of functional data has recently received increasing attention thanks to the flexibility of permutation tests to handle complex testing problems. The purpose of this work is to present some new insights in the context of nonparametric inference on functional data using the permutation approach, more specifically we formally show the equivalence of some permutation procedures proposed in the literature and we suggest the use of the permutation and combination-based approach within the basis function approximation layout. Validation of theoretical results is shown by simulation studies.