Wall-crossing integral Chow rings of ${\bar{\mathcal{M}}}_{1,n}$

We compute the integral Chow rings of ${\bar{\mathcal{M}}}_{1,n}$ for $n=3,4$. For $n\le 6$, these stacks can be obtained by a sequence of weighted blow-ups and blow-downs from a simple stack, either a weighted projective space or a Grassmannian. Our strategy consists in inductively computing all the integral Chow rings of the alternative compactifications introduced by Smyth and studied by Lekili-Polishchuk.