We report the evolution of thermal transport properties of iron-based superconductor K$_x$Fe$_{2-y}$Se$_2$ with sulfur substitution at Se sites. Sulfur doping suppresses the superconducting $T_c$ as well as the Seebeck coefficient. The Seebeck coefficient of all crystals in the low temperature range can be described very well by diffusive thermoelectric response model. The zero-temperature extrapolated value of Seebeck coefficient divided by temperature $S/T$ gradually decreases from $-0.48 mu V/K^2$ to a very small value $sim$ 0.03 $mu$V/K$^2$ where $T_c$ is completely suppressed. The normal state electron Sommerfeld term ($gamma_n$) of specific heat also decreases with the increase of sulfur content. The dcrease of $S/T$ and $gamma_n$ reflects a suppression of the density of states at the Fermi energy, or a change in the Fermi surface that would induce the suppression of correlation strength.
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