  We provide a moment map interpretation for the coupled K\"ahler-Einstein
equations introduced by Hultgren and Witt Nystr\"om, and in the process
introduce a more general system of equations, which we call coupled cscK
equations. A differentio-geometric formulation of the corresponding Futaki
invariant is obtained and a notion of K-polystability is defined for this new
system. Finally, motivated by a result of Sz\'ekelyhidi, we prove that if there
is a solution to our equations, then small K-polystable perturbations of the
underlying complex structure and polarizations also admit coupled cscK metrics.
