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superseded by antiferromagnetic collinear long-range order (k=0) at low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