In this paper, we focus on understanding the physical nature of continuum based modelling of size effects in small size samples. The differences between the classical, higher-grade and nonlocal continuum models are pointed out. In the higher-grade theories, the order of derivatives in the governing partial differential equations is increased and C0 continuity approximation of field variables becomes insufficient. There are proposed two alternatives, how to achieve the demand of high continuity without increasing the amount of degrees of freedom. The Moving Finite Element Approximation is implemented into meshless formulations, while the Mixed FEM is used in global FEM formulations. Finally, the stationary heat conduction problem in an infinitely extended bilayer is considered within the higher-grade continuum theory. Making use the analytical solution, there are analyzed necessary conditions under which the temperature is distributed nonlinearly throughout the bilayer thickness and size-effects are observed.