A systematic study of the structural and magnetic properties of three-dimensionally frustrated lanthanide garnets $Ln_3A_2X_3text{O}_{12}$, $Ln$ = Gd, Tb, Dy, Ho, $A$ = Ga, Sc, In, Te, $X$ = Ga, Al, Li is presented. Garnets with $Ln$ = Gd show magnetic behaviour consistent with isotropic Gd$^{3+}$ spins; no magnetic ordering is observed for T $geq$ 0.4 K. Magnetic ordering features are seen for garnets with $Ln$ = Tb, Dy, Ho in the temperature range 0.4 < T < 2.5 K, however the nature of the magnetic ordering varies for the different $Ln$ as well as for different combinations of $A$ and $X$. The changes in magnetic behaviour can be explained by tuning of the magnetic interactions and changes in the single-ion anisotropy. The change in magnetic entropy is evaluated from isothermal magnetisation measurements to characterise the magnetocaloric effect in these materials. Among the Gd garnets, the maximum change in magnetic entropy per mole (15.45 J K$^{-1}$ mol$_{text{Gd}}^{-1}$) is observed for Gd$_3$Sc$_2$Ga$_3$O$_{12}$ at 2 K, in a field of 9 T. The performance of Dy$_3$Ga$_5$O$_{12}$ as a magnetocaloric material surpasses the other garnets with $Ln$ = Tb, Dy, Ho.
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