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.