In this paper we report on a study, using neutron scattering techniques, of the spin fluctuations in the random magnetic-nonmagnetic antiferromagnets,

{Mn}_{c} {Zn}_{1 - c} F_{2}

and

K {Mn}_{c} {Zn}_{1 - c} F_{3}

, with

c

close to the percolation concentration

c_{p}

. The results for

c < c_{p}

are consistent with the multicritical point picture for the percolation point with the temperature scale determined by the inverse correlation length of the appropriate linear chain. This approach gives a very satisfactory account of the spin-space crossover from Heisenberg-like to Ising-like behavior observed in tetragonal

{Mn}_{c} {Zn}_{1 - c} F_{2}

. The exponent

ν_{T}

describing the temperature dependence of the inverse correlation length is found to be 0.85 ± 0.10 for this system and, with rather less certainty, we infer

ν_{T} = 0.95 \pm 0.05

for the cubic

K {Mn}_{c} {Zn}_{1 - c} F_{3}

. The results for

c > c_{p}

show two unexpected features. Firstly the diffuse scattering does not show any critical scattering associated with the onset of long-range order. We suggest this may be because only a small fraction of spins are in the backbone of the infinite cluster. Secondly in

K {Mn}_{c} {Zn}_{1 - c} F_{3}

the long-range order decreases on cooling below 6 K. We suggest that this may result from the random magnetic dipole-dipole forces producing a type of spin-glass phase.

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