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AlScN is a new wide bandgap, high-k, ferroelectric material for RF, memory, and power applications. Successful integration of high quality AlScN with GaN in epitaxial layer stacks depends strongly on the ability to control lattice parameters and surface or interface through growth. This study investigates the molecular beam epitaxy growth and transport properties of AlScN/GaN multilayer heterostructures. Single layer Al$_{1-x}$Sc$_x$N/GaN heterostructures exhibited lattice-matched composition within $x$ = 0.09 -- 0.11 using substrate (thermocouple) growth temperatures between 330 $ ^\circ$C and 630 $ ^\circ$C. By targeting the lattice-matched Sc composition, pseudomorphic AlScN/GaN multilayer structures with ten and twenty periods were achieved, exhibiting excellent structural and interface properties as confirmed by X-ray diffraction (XRD) and scanning transmission electron microscopy (STEM). These multilayer heterostructures exhibited substantial polarization-induced net mobile charge densities of up to 8.24 $\times$ 10$^{14}$/cm$^2$ for twenty channels. The sheet density scales with the number of AlScN/GaN periods. By identifying lattice-matched growth condition and using it to generate multiple conductive channels, this work enhances our understanding of the AlScN/GaN material platform.