Fusion of two conformal defects in conformal field theory can be understood as an RG flow whose IR fixed point is another conformal defect, with the running scale is set by the separation between the defects. When the separation is small, the system can be described by EFT techniques, in terms of an effective action on the IR defect. In this talk I will discuss the constraints that conformal symmetry imposes on such effective actions, and the implications of this picture for observables such as the cusp anomalous dimension. Joint work with Alexander Radcliffe and Ritam Sinha, arXiv:2406.04561.