Let $ f:X\rightarrow Y$ be a proper morphism of smooth Deligne-Mumford stacks of finite type over $ \mathbb{C}$ that is birational, but not flat. The coarse spaces of $ X$ and $ Y$ are both not smooth. Is proper pushforward $ f_*:A^*(X)_{\mathbb{Q}}\rightarrow A^*(Y)_{\mathbb{Q}}$ is surjective?

The way I want to prove a statement like this is using a projection formula argument on the coarse spaces, but I don’t have a reasonable way to define pullback in the absence of flatness and the fact that the coarse spaces aren’t smooth. Another idea is to take a cycle on $ Y$ and try to move it out of the exceptional locus, take its preimage, and then push that forward, but I don’t know if there is a moving lemma on stacks. I can’t apply it on the coarse space because the coarse space of $ Y$ is not regular. Can I remedy one of these arguments by working on the level of the stack instead of the coarse space?