We report the successful substitution of cobalt, nickel, and copper for iron in the 1038 phase parent compound Ca$_{10}$(FeAs)$_{10}$(Pt$_3$As$_8$) yielding Ca$_{10}$(Fe$_{1-x}$Co$_x$As)$_{10}$(Pt$_3$As$_8$), Ca$_{10}$(Fe$_{1-x}$Ni$_x$As)$_{10}$(Pt$_3$As$_8$), and Ca$_{10}$(Fe$_{1-x}$Cu$_x$As)$_{10}$(Pt$_3$As$_8$), respectively. Superconductivity is induced in Co and Ni doped compounds reaching critical temperatures up to 15 K, similar to known Pt substituted Ca$_{10}$(Fe$_{1-x}$Pt$_x$As)$_{10}$(Pt$_3$As$_8$), whereas no superconductivity was detected in Ca$_{10}$(Fe$_{1-x}$Cu$_x$As)$_{10}$(Pt$_3$As$_8$). The obtained Tc(x) phase diagrams are very similar to those of other iron arsenide superconductors indicating rather universal behavior despite the more complex structures of the 1038-type compounds, where the physics is primarily determined by the FeAs layer.