  We present a comprehensive study of the low-temperature heat capacity and
thermal expansion of single crystals of the hole-doped Ba1-xKxFe2As2 series
(0<x<1) and the end-members RbFe2As2 and CsFe2As2. A large increase of the
Sommerfeld coefficient is observed with both decreasing band filling and
isovalent substitution (K, Rb, Cs) revealing a strong enhancement of electron
correlations and the possible proximity of these materials to a Mott insulator.
This trend is well reproduced theoretically by our Density-Functional Theory +
Slave-Spin (DFT+SS) calculations, confirming that 122-iron pnictides are
effectively Hund metals, in which sizable Hund's coupling and orbital
selectivity are the key ingredients for tuning correlations. We also find
direct evidence for the existence of a coherence-incoherence crossover between
a low-temperature heavy Fermi liquid and a highly incoherent high-temperature
regime similar to heavy fermion systems. In the superconducting state, clear
signatures of multiband superconductivity are observed with no evidence for
nodes in the energy gaps, ruling out the existence of a doping-induced change
of symmetry (from s to d-wave). We argue that the disappearance of the electron
band in the range 0.4<x<1.0 is accompanied by a strong-to-weak coupling
crossover and that this shallow band remains involved in the superconducting
pairing, although its contribution to the normal state fades away. Differences
between hole- and electron-doped BaFe2As2 series are emphasized and discussed
in terms of strong pair breaking by potential scatterers beyond the Born limit.
