Summary: For any reductive group \(G\) and a parabolic subgroup \(P\) with its Levi subgroup \(L\), the first author introduced in the first part [Math. Ann. 366, No. 1–2, 395–415 (2016; Zbl 1359.14044)] a ring homomorphism \(\xi_\lambda^P : \operatorname{Rep}_{\lambda - \operatorname{poly}}^{\mathbb{C}}(L) \to H^\ast(G / P, \mathbb{C})\), where \(\operatorname{Rep}_{\lambda - \operatorname{poly}}^{\mathbb{C}}(L)\) is a certain subring of the complexified representation ring of \(L\) (depending upon the choice of an irreducible representation \(V(\lambda)\) of \(G\) with highest weight \(\lambda\)). In this paper we study this homomorphism for \(G = \operatorname{Sp}(2 n)\) and its maximal parabolic subgroups \(P_{n - k}\) for any \(1 \leq k \leq n - 1\) (with the choice of \(V(\lambda)\) to be the defining representation \(V( \omega_1)\) in \(\mathbb{C}^{2 n}\)). Thus, we obtain a \(\mathbb{C} \)-algebra homomorphism \(\xi_{n, k} : \operatorname{Rep}_{\omega_1 - \operatorname{poly}}^{\mathbb{C}}(\operatorname{Sp}(2 k)) \to H^\ast(I G(n - k, 2 n), \mathbb{C})\). Our main result asserts that \(\xi_{n, k}\) is injective when \(n\) tends to \(\infty\) keeping \(k\) fixed. Similar results are obtained for the odd orthogonal groups.