Consider two independent random variables $X$ and $Y$. Let $\mathbb{Y}$ be the support of $Y$ and take a function $f:\mathbb{Y}\rightarrow \mathbb{R}$. Consider the distribution of $$ X+f(Y) \text{ conditional on $Y=y$} $$ and the distribution of $$ X+f(Y) \text{ conditional on $Y=y'$} $$ Is it correct to say that these two distributions will have all equal moments except the first one?

This is correct when $X$ has, for instance, a Normal distribution. However, I am not sure if it holds generically.