Posting YCor's comment to hopefully end some confusion -
of course details are needed. But it seems that Fubini applied to $|f(gx)-f(x)|$ on $G\times X$ implies that for a.e. all $x$ we have $f(gx)=f(x)$ for a.e. all $g$. Since for given $x$ and every measure-generic subset $U$ of $G$ we have $Ux$ measure-generic in $X$ (for $G$ second-countable and $X$ being $G$ mod discrete this seems quite clear), it follows that $f$ is a.e. constant.