For an arbitrary operator $T$ on Hilbert space, we study the maps $\widetilde{\Phi }:f(T)\rightarrow f(\widetilde{T})$ and $\widehat{\Phi } :f(T)\rightarrow f(\widehat{T}),$ where $\widetilde{T}$ and $\widehat{T}$ are the Aluthge and Duggal transforms of $T$, respectively, and $f$ belongs to the algebra \textrm{Hol}$(\sigma (T))$. We show that both maps are (contractive and) completely contractive algebra homomorphisms. As applications we obtain that every spectral set for $T$ is also a spectral set for $\widehat{T}$ and $\widetilde{T}$, and also the inclusion $W(f( \widetilde{T}))^{-}\cup W(f(\widehat{T}))^{-}\subset W(f(T))^{-}$ relating the numerical ranges of $f(T),$ $f(\widetilde{T}),$ and $f(\widehat{T}).$