In the framework of the general phase diagram published recently for a pseudospin S = 1/2 on a hyperkagome network [Kibalin et al., Phys. Rev. Res. 2, 033509 (2020)], four Dy₃B₂C₃O₁₂ garnets with different species on the B and C sites, namely, Dy₃Sc₂Al₃O₁₂ (DyScAl), Dy₃Te₂Li₃O₁₂ (DyTeLi), Dy₃Sc₂Ga₃O₁₂ (DyScGa), and Dy₃In₂Ga₃O₁₂ (DyInGa), have been studied by neutron powder diffraction and inelastic neutron scattering, to investigate the impact of B and C substitutions on the Dy³⁺ dodecahedral oxygen environment and crystal field, and thus on the Dy³⁺ spin magnetic anisotropy. In the four compounds, the magnetic ground state is a multiaxis antiferromagnetic order similar to that of Dy₃Al₅O₁₂, characterized by an ordered moment close to the theoretical maximum, ∼6µ_B. The gap between the ground state and the first excited crystal electric field level varies from ∼3.2 meV (DyInGa) up to 11.2 meV (DyScAl), confirming the rather strong Ising character of the Dy³⁺ moment in those garnets. For a better understanding of the impact of the distortion of the oxygen environment on the Dy³⁺ magnetic anisotropy, point-charge calculations were then performed, mapping a magnetic anisotropy diagram versus a wide range of distortions. Results show that Ising-like behaviors mostly predominate, associated with large g_x, g_y, or g_z Landé values, whose stable area is determined by the shape of the DyO₈ cage. Magnetic entropy calculations confirm that the magnetocaloric effect increases sharply near those boundaries and validate the use of dysprosium garnets for low temperature, low magnetic field adiabatic demagnetization refrigeration cryostats.