The non-identical particle femtoscopy is a technique that is developed to estimate the dimension of a particle-emitting medium as well as the average pair-emission asymmetry between the particles using two-particle correlation functions. The analysis of femtoscopic correlations for all possible charged combinations of pion-kaon pairs in Pb$-$Pb collisions at $\sqrt{s_{\mathrm{NN}}}=$ 5.02 TeV with ALICE at the LHC is presented in this article. The results are extracted for different centrality classes of Pb$-$Pb collisions using the spherical harmonics representations of the correlation functions, $C^{\rm 0}_{\rm 0}$ and $Re C^{\rm 1}_{\rm 1}$. The evolution of the femtoscopic parameters with multiplicity and collectivity of the system is discussed extensively. The femtoscopic parameters are observed to increase with the multiplicity. The non-zero values of pair-emission asymmetry and the decrease of source-size with increasing pair-transverse momentum indicate the presence of collectivity in the system.