Let $f:X\rightarrow Y$ be a morphism where $X$ and $Y$ are smooth projective varieties over $\mathbb{C}$. Does $f$ induce a well-defined "pushforward" morphism $\overline{\mathcal{M}}_{g,n}(X,\beta)\rightarrow \overline{\mathcal{M}}_{g,n}(Y,f_*\beta)$ in the category of DM stacks?
I know the answer is yes if $Y$ is a point. In this case, it is called the stablization morphism. I also know the tricky step for defining it is to contract all bubbles which are no longer stable after forgetting the target of a given stable map.
By the same argument, I am thus convinced that the "pushforward" morphism does exist set-theoretically. But how do we prove that it is a morphism in the category of DM stacks?