The magnetic order and the spin dynamics in the antiferromagnetic entropy-stabilized oxide (Mg$_{0.2}$Co$_{0.2}$Ni$_{0.2}$Cu$_{0.2}$Zn$_{0.2}$)O (MgO-ESO) have been studied using muon spin relaxation ($mu$SR) and inelastic neutron scattering. We find that antiferromagnetic order develops gradually in the sample volume as it is cooled below 140 K, becoming fully ordered around 100 K. The spin dynamics show a critical slowing down in the vicinity of the transition, and the magnetic order parameter grows continuously in the ordered state. These results indicate that the antiferromagnetic transition is continuous but proceeds with a Gaussian distribution of ordering temperatures. The magnetic contribution to the specific heat determined from inelastic neutron scattering likewise shows a broad feature centered around 120 K. High-resolution inelastic neutron scattering further reveals an initially gapped spectrum at low temperature which sees an increase in a quasielastic contribution upon heating until the ordering temperature.
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