Isomorphism of Principal Bundles with structure groupoid

Question

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid. Suppose that $\pi:P\rightarrow B$ is a $\mathcal{G}$-principal bundle and let $(h_s,g_s):(P,B)\rightarrow (P,B); s\in [0,1]$, be a homotopy of $\mathcal{G}$-principal bundle morphisms, that is:

for each $s\in [0,1]$ we have $h_s:P\rightarrow P$, $\mathcal{G}$-equivariant map and $g_s:B\rightarrow B$, such that $\pi\circ h_s=g_s\circ \pi$, in such a way that the families $(h_s)_{s\in [0,1]}$ and $(g_s)_{s\in [0,1]}$ are homotopies.

I want to prove that the $\mathcal{G}$-bundles $g_0^{\ast}(P)\rightarrow B$ and $g_1^{\ast}(P)\rightarrow B$ are isomorphic to each other. Is that true? Can anyone give a proof or a reference please?