In this talk, I will introduce vertical configuration spaces in R^{p,q}, a generalization of configuration spaces with additional vertical coordinate conditions. These spaces admit a natural action by (a product of ) wreath products. As the number of points in a configuration increases, while the (co)homology groups of these spaces grow increasingly complex, their patterns as wreath product representations stabilize. This phenomenon, known as representation stability, was first introduced by Church and Farb in 2010. I will explain the underlying algebraic structure responsible for this stability, known as FI-modules, developed by Church, Ellenberg, and Farb. I will then introduce a decorated version of this framework for wreath products, called FI_G-modules, and illustrate how the sequence of rational (co)homology groups of the vertical configuration spaces exhibits representation stability. This talk is based on joint work with David Baron, Urshita Pal, Jenny Wilson, and Chunye Yang.