  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). In this paper, we
give a condition for a weighted soliton on a Fano manifold $M_0$ to extend to
weighted solitons on small deformations $M_t$ of the Fano manifold $M_0$. More
precisely, 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. This generalizes
a result of Cao-Sun-Yau-Zhang for Kaehler-Einstein metrics.
