  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.
