Single crystals of Ba${}_{1\ensuremath{-}x}$Eu${}_{x}$Fe${}_{2}$As${}_{2}$ were studied by magnetic susceptibility, heat capacity, resistivity, and electron spin resonance (ESR) measurements. Spin-density wave (at ${T}_{\text{SDW}}$) and antiferromagnetic (at ${T}_{N}$) phase transitions were mapped as a function of $x$. For $x\ensuremath{\ge}0.2$, we found a single Eu${}^{2+}$ ESR Dysonian line that presents an isotropic linear increase (Korringa) of its linewidth ($\ensuremath{\Delta}H$) above ${T}_{\text{SDW}}$ which systematically decreases with decreasing $x$. In contrast, for a critical concentration ${x}_{c}$ ($0.10l{x}_{c}l0.20$), $\ensuremath{\Delta}H$ decreases with increasing $T$, suggesting a distinct relaxation process that we associate with a Eu${}^{2+}$ Kondo single impurity regime. The Korringa rate suppression towards the Ba-rich compounds is claimed to be due to the reduction of the $q$-dependent exchange interaction between the Eu${}^{2+}$ $f$ electrons and the conduction electrons, which is likely associated with an increasing of localization of Fe $d$ electrons. This result may help the understanding of the SDW phase suppression (that can lead to superconductivity) in this class of materials.