Pullbacks of stacks coming from Lie groupoids are not always equivalent to Lie groupoids.
Take $G=H=\mathbb{R}$. Define $F(x)=0$ if $x\leq 0$ and $F(x)=exp(−1/x^2)$ if $x>0$.
The pullback is not equivalent to a Lie groupoid in this situation: the set-theoretical pullback is $(−\infty,0]\times(−\infty,0]\cup \{(x,x)|x\in \mathbb{R}\}$, which is clearly not a smooth manifold.
One can guarantee that the pullback is a Lie groupoid by imposing transversality conditions on the maps involved. That is, $A \times_C B$ is a Lie groupoid if $A_0 \rightarrow C_0 \leftarrow B_0$ is transversal and $A_1 \rightarrow C_1 \leftarrow B_1$ is transversal, where subscripts $0$ and $1$ denotes objects and morphisms respectively. (In fact, this transversality condition guarantees that the pullback is also a homotopy pullback, which is almost always what one actually wants.)
