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The present study aims at estimating the heat and the mass transfer coefficients in the case of the polyatomic gas flows through long rectangular microchannels driven by small and large pressure (Poiseuille flow) and temperature (Thermal creep flow) drops. The heat and mass transfer coefficients are presented for all gas flow regimes, from free molecular up to hydrodynamic ones, and for channels with different aspect ratios as well as for various values of translational and rotational Eucken factors. The applied values of the Eucken factors were extracted based on the Rayleigh-Brillouin experiments and the kinetic theory of gases. The numerical study has been performed on the basis of a kinetic model for linear and non-linear gas molecules considering the translational and rotational degrees of freedom. The solution of the obtained system of the kinetic equations is implemented on the Graphics Processing Units (GPUs), allowing the reduction of the computational time by two orders of magnitude. The results show that the Poiseuille mass transfer coefficient is not affected by the internal degrees of freedom and the non-dependence of the previous observed deviations with the experimental data on the molecular nature of the gas molecules is confirmed. However, the study shows that the deviation between monatomic and polyatomic values of the mass transfer coefficient in the thermal creep flow is increased as the gas rarefaction is decreased, and for several polyatomic gases met in practical applications in the temperature range from 300 to 900 K might reach 15%. In addition, the effect of the internal degrees of freedom on the heat transfer coefficient is found to be rather significant. The polyatomic heat transfer coefficients are obtained essentially higher than the monatomic ones, with the maximum difference reaching about 44% and 67% for linear and non-linear gas molecules. In view of the large differences between monatomic and polyatomic gases, the present results may be useful in the design of technological devices in which the thermal creep phenomenon plays a dominant role.