We consider weighted solitons on Fano manifolds which include Kaehler-Ricci solitons, Mabuchi solitons and base metrics which induce Calabi-Yau cone metrics outside the zero sections of the canonical line bundles (Sasaki-Einstein metrics on the associated U(1)-bundles). We show that all the members M_t of the Kuranishi family of a Fano manifold M_0 with a weighted soliton have weighted solitons if and only if the dimensions of T-equivariant automorphism groups of M_t are equal to that of M_0, and also if and only if the T-equivariant automorphism groups of M_t are all isomorphic to that of M_0, where the weight functions are defined on the moment polytope of the Hamiltonian T-action.