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We report on systematic muon-spin rotation and relaxation ($\mu$SR) studiesof the magnetic properties of EuAl$_4$ and EuGa$_4$ single crystals at amicroscopic level. Transverse-field $\mu$SR measurements, spanning a widetemperature range (from 1.5 to 50 K), show clear bulk AFM transitions, with analmost 100% magnetic volume fraction in both cases. Zero-field $\mu$SRmeasurements, covering both the AFM and the paramagnetic (PM) states, revealinternal magnetic fields $B_\mathrm{int}(0) = 0.33$ T and 0.89 T in EuAl$_4$and EuGa$_4$, respectively. The transverse muon-spin relaxation rate$\lambda_\mathrm{T}$, a measure of the internal field distribution at themuon-stopping site, shows a contrasting behavior. In EuGa$_4$, it decreaseswith lowering the temperature, reaching its minimum at zero temperature,$\lambda_\mathrm{T}(0) = 0.71$ $\mu$s$^{-1}$. In EuAl$_4$, it increasessignificantly below $T_\mathrm{N}$, to reach 58 $\mu$s$^{-1}$ at 1.5 K, mostlikely reflecting the complex magnetic structure and the competing interactionsin the AFM state of EuAl$_4$. In both compounds, the temperature-dependentlongitudinal muon-spin relaxation $\lambda_\mathrm{L}(T)$, an indication of therate of spin fluctuations, diverges near the onset of AFM order, followed by asignificant drop at $T < T_\mathrm{N}$. In the AFM state, spin fluctuations aremuch stronger in EuAl$_4$ than in EuGa$_4$, while being comparable in the PMstate. The evidence of robust spin fluctuations against the external magneticfields provided by $\mu$SR may offer new insights into the origin of thetopological Hall effect and the possible magnetic skyrmions in the EuAl$_4$ andEuGa$_4$ compounds.

Spin order and fluctuations in the EuAl$_4$ and EuGa$_4$ topological antiferromagnets: A $\mu$SR study