To prevent mechanical stress in a film-substrate system, there is suggested to grow up thin layers of a non-magnetic garnet Gd$_3$Ga$_5$O$_{12}$ (Y:GGG) (0 $\leq$ х $\leq$ 0.2) on a gadolinium gallium garnet Gd$_{3-x}$Y$_x$Ga$_5$O$_{12}$ (GGG) ($a_s$ = 12.383 Å) substrate. These thin layers have a negative gradient of a crystal lattice parameter towards on yttrium iron garnet Y$_3$Fe$_5$O$_{12}$ (YIG) ($a_f$ = 12.376 Å) film, that is grown using liquid phase epitaxy method on the transitional layer. By regulating the technological conditions of growing (temperature and growth rate, solution-melt overcooling degree) the transitional layers Y:GGG in the thickness from 2.2 to 6.4 $\mu$m are received. Their crystal lattice parameter smoothly varies from a value of 12.3827 Å near the substrate with GGG to 12.3762–12.3789 Å on the surface of the layer. The YIG films grown up on the substrates with transitional layer have a thickness up to 130 $\mu$m, their ferromagnetic resonance line has a $\Delta H$ = (0.34–0.57) Oe width with a thickness $\leq$73 $\mu$m.