[Additional explanation]

Let the circle group $S^1$ act on the unit sphere $S^2 \subset \mathbb{R}^3$ as rotations about the $z$-axis. For the action groupoid $\mathcal{G}$ of the action, the space $(\mathcal{G}^1)_0=\lbrace (t,x) \in S^1 \times S^2 \mid\ t=1 \text{ if } x \ne (0,0,\pm 1) \rbrace$. This is not a manifold. So if the space $(\mathcal{G}^k)_0$ is to be a manifold, then we have to assume some conditions. (This is the reason why I have changed the title of this question.)

I have found a relavant explanation in Moerdijk. In section 6.4, he deals with the inertia orbifold (groupoid). According to the paper, if $\mathcal{G}$ is étale, then we can show by taking local chats that $S_\mathcal{G} (=(\mathcal{G}^1)_0)$ is a manifold.

Moreover for a proper foliation groupoid, we can also use "local charts" in the sense of Crainic-Moerdijk to show that the smoothness of $S_\mathcal{G}$. So I am checking Crainic-Moerdijk.

But the problem still remains even if we understand that $(\mathcal{G}^1)_0$ is a manifold. Let $\pi_k:(\mathcal{G}^k)_0 \to G_0$ is the map sending $(a_1,\dots)$ to $s(a_1)$. Then $(\mathcal{G}^{k+1})_0$ is the fiber product of $\pi_k$ and $\pi_1$ as a topological space. But $\pi_1$ is not a submersion. In fact, let the circle group acts on the unit 3-sphere in $\mathbb{C}^2$ with multiplicity $(1,p)$. (The quotient is the orbifold called a tear drop.) Then $(\mathcal{G}^1)_0$ consists of the original 3-sphere with $p-1$ circles. Therefore $\pi_1$ is not a submersion.

Let ${\mathcal{G} = \lbrace s,t:G_1 \to G_0 \rbrace}$ be a Lie groupoid. Define $$(\mathcal{G}^k)_0:=\lbrace (a_1,\dots,a_k) \in G_1^k\mid s(a_1)=t(a_1)=\dots=s(a_k)=t(a_k) \rbrace$$ (This is the space of objects of $k$-sectors $\mathcal{G}^k$. See Adem-Ruan-Zhang [arXiv:math/0605534] arXiv:math/0605534 for more details.)

My question is: how do we prove that the space $(\mathcal{G}^k)_0$ is a manifold?

(Or how do we see that the map $$ G_1^k \to G_0^{2k}: (a_1,\dots,a_k)\mapsto(s(a_1),t(a_1),\dots,s(a_k),t(a_k)) $$ is transverse to $ Z=\lbrace(x,\dots,x) \in G_0^{2k}\mid x \in G_0 \rbrace $ ?)

Let ${\mathcal{G} = \lbrace s,t:G_1 \to G_0 \rbrace}$ be a Lie groupoid. Define $$(\mathcal{G}^k)_0:=\lbrace (a_1,\dots,a_k) \in G_1^k\mid s(a_1)=t(a_1)=\dots=s(a_k)=t(a_k) \rbrace$$ (This is the space of objects of $k$-sectors $\mathcal{G}^k$. See Adem-Ruan-Zhang [arXiv:math/0605534] for more details.)

My question is: how do we prove that the space $(\mathcal{G}^k)_0$ is a manifold?

(Or how do we see that the map $$ G_1^k \to G_0^{2k}:(a_1,\dots,a_k)\mapsto(s(a_1),t(a_1),\dots,s(a_k),t(a_k))$ G_0^{2k}: (a_1,\dots,a_k)\mapsto(s(a_1),t(a_1),\dots,s(a_k),t(a_k)) $$ is transverse to $ Z=\lbrace(x,\dots,x) Z=\lbrace(x,\dots,x) \in G_0^{2k}\mid x \in G_0 \rbrace$?)rbrace $ ?)

Let ${\mathcal{G} = \lbrace s,t:G_1 \to G_0 \rbrace}$ be a Lie groupoid. Define $$(\mathcal{G}^k)_0:=\lbrace (a_1,\dots,a_k) \in G_1^k\mid s(a_1)=t(a_1)=\dots=s(a_k)=t(a_k) \rbrace$$ (This is the space of objects of $k$-sectors $\mathcal{G}^k$. See Adem-Ruan-Zhang [arXiv:math/0605534] for more details.)

My question is: how do we prove that the space $(\mathcal{G}^k)_0$ is a manifold?

(Or how do we see that the map $G_1^k \to G_0^{2k}:(a_1,\dots,a_k)\mapsto(s(a_1),t(a_1),\dots,s(a_k),t(a_k))$ is transverse to $Z=\lbrace(x,\dots,x) \in G_0^{2k}\mid x \in G_0 \rbrace$?)