Affine W-algebras form a family of vertex algebras indexed by the nilpotent orbits of a simple finite dimensional complex Lie algebra. Each of them is built as a noncommutative Hamiltonian reduction of the corresponding affine Kac-Moody algebra. In this talk, I will present a joint work with Naoki Genra about the problem of reduction by stages for these affine W-algebras: given a suitable pair of nilpotent orbits in the simple Lie algebra, it is possible to reconstruct one of the two affine W-algebras associated to these orbits as the Hamiltonian reduction of the other one. I will insist on how this problem relates to our previous work about reduction by stages between Slodowy slices, which are Poisson varieties associated with affine W-algebras. I will also mention some applications and motivations coming from the Kraft-Procesi rule for nilpotent Slodowy slices, and isomorphisms between simple affine admissible W-algebras.