We give a Morita equivalence theorem for cyclotomic Hecke algebras of type B and D, in the spirit of a classical result of Dipper–Mathas in type A for Ariki–Koike algebras. The main step in the proof consists in a decomposition theorem for generalisations of quiver Hecke algebras that appeared recently in the study of affine Hecke algebras of type B and D. This theorem reduces the general situation of a disconnected quiver with involution to a simpler setting. To be able to treat types B and D at the same time we unify the different definitions of generalisations of quiver Hecke algebra for type B that exist in the literature, and study their structural properties.