  We discuss the results of $^{75}$As Nuclear Quadrupole Resonance (NQR) and
muon spin relaxation measurements in AFe$_2$As$_2$ (A= Cs, Rb) iron-based
superconductors. We demonstrate that the crossover detected in the nuclear
spin-lattice relaxation rate $1/T_1$ (around 150 K in RbFe$_2$As$_2$ and around
75 K in CsFe$_2$As$_2$), from a high temperature nearly localized to a low
temperature delocalized behaviour, is associated with the onset of an
inhomogeneous local charge distribution causing the broadening or even the
splitting of the NQR spectra as well as an increase in the muon spin relaxation
rate. We argue that this crossover, occurring at temperatures well above the
phase transition to the nematic long-range order, is associated with a charge
disproportionation at the Fe sites induced by competing Hund's and Coulomb
couplings. In RbFe$_2$As$_2$ around 35 K, far below that crossover temperature,
we observe a peak in the NQR $1/T_1$ which is possibly associated with the
critical slowing down of electronic nematic fluctuations on approaching the
transition to the nematic long-range order.
