Finite W-algebras are a finite collection of filtered algebras associated to each complex semisimple Lie algebra, which have interesting applications to the classification of primitive ideals in enveloping algebras. One of the key challenges in the theory is to find an explicit presentation for a finite W-algebra. This problem was solved comprehensively for the general linear algebras by Brundan--Kleshchev, by relating them to shifted Yangians, with further important work by Kac--De Sole and their collaborators. Extending this to other classical Lie algebras has proved to be extremely difficult. A good approximation to the problem is describing the Poisson structure on the semiclassical limit of the W-algebra. In this seminar I will describe some new progress in extending the Yangian description to types B, C, D in the semiclassical setting, using the theory of Dirac reduction.