We consider the dynamics of dark matter solitons moving through non-uniform cigar-shaped Bose-Einstein condensates described by the mean field Gross-Pitaevskii equation with generalized nonlinearities, in the case when the condition for the modulation stability of the Bose-Einstein condensate is fulfilled. The analytical expression for the frequency of the oscillations of a deep dark soliton is derived for nonlinearities which are arbitrary functions of the density, while specific results are discussed for the physically relevant case of a cubic-quintic nonlinearity modeling two- and three-body interactions, respectively. In contrast to the cubic Gross-Pitaevskii equation for which the frequencies of the oscillations are known to be independent of background density and interaction strengths, we find that in the presence of a cubic-quintic nonlinearity an explicit dependence of the oscillations frequency on the above quantities appears. This dependence gives rise to the possibility of measuring these quantities directly from the dark soliton dynamics, or to manage the oscillation via the changes of the scattering lengths by means of Feshbach resonance. A comparison between analytical results and direct numerical simulations of the cubic-quintic Gross-Pitaevskii equation shows good agreement which confirms the validity of our approach.
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