Equivariant Poisson-Lie T-duality is a wide extension of the standard PLT duality. Both constructions were introduced by Klimčík and Ševera in the 90's, the former under then name 'dressing cosets'. We will discuss how to formulate this extension in terms of Courant algebroid reductions and show that this duality is compatible with the string background equations. If time permits, we will also discuss type II supergravity and an algebraic method for searching new solutions of background equations. This is a joint work with Pavol Ševera.