Using neutron powder diffraction and Monte-Carlo simulations we show that a spin-liquid regime emerges at $all compositions in the diamond-lattice antiferromagnets Co(Al1-xCox)2O4. This spin-liquid regime induced by frustration due to the second-neighbour exchange coupling J2, is gradually superseded by antiferromagnetic collinear long-range order (k=0) at low temperatures. Upon substitution of Al3+ by Co3+ in the octahedral B-site the temperature range occupied by the spin-liquid regime narrows and TN increases. To explain the experimental observations we considered magnetic anisotropy D or third-neighbour exchange coupling J3 as degeneracy-breaking perturbations. We conclude that Co(Al1-xCox)2O4 is below the theoretical critical point J2/J1=1/8, and that magnetic anisotropy assists in selecting a collinear long-range ordered ground state, which becomes more stable with increasing x due to a higher efficiency of O-Co3+-O as an interaction path compared to O-Al3+-O.
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