Muon pairs created in the Drell-Yan process of a negative pion beam of 190 GeV/c momentum impinging on a transversely polarised NH$_3$ target were detected with the COMPASS spectrometer at CERN in 2015. In addition to the extraction of the transverse spin asymmetries (TSAs) from these data, a complementary analysis of the TSAs weighted by powers of the dimuon momentum $q_T$ has been performed and is presented in this paper. In the transverse momentum dependent parton distribution functions (TMD PDFs) formalism, the $q_T$-weighted TSAs can be interpreted in terms of products of the TMD PDFs of the beam pion and of the transversely polarised target proton, unlike the conventional TSAs, which are interpreted as their convolutions. This allowed us to make a straightforward comparison with the expectation based on the weighted Sivers asymmetry measured in the SIDIS process. In a similar way, the Boer-Mulders function of the pion and proton can be obtained as well.