Suppose $[X_1\rightrightarrows X_0]$ and $[Y_1\rightrightarrows Y_0]$ are Morita equivalent Lie groupoids. This means, there exists another Lie groupoid $[Z_1\rightrightarrows Z_0]$ and Morita morphisms $[Z_1\rightrightarrows Z_0]\rightarrow [X_1\rightrightarrows X_0]$ and $[Z_1\rightrightarrows Z_0]\rightarrow [Y_1\rightrightarrows Y_0]$.

Another alternative description is, there exists a $[X_1\rightrightarrows X_0]-[Y_1\rightrightarrows Y_0]$ ``biprincipal bibundle" $X_0\xleftarrow{\phi} P\xrightarrow{\varphi} Y_0$.

Does it imply some specific condition on the maps $\varphi$ and $\phi$?

For example, can the differentials $\phi_{*,a}:T_aP\rightarrow T_{\phi(a)}X_0$ and $\varphi_{*,a}:T_aP\rightarrow T_{\varphi(a)}Y_0$ have some common properties for $a\in P?$ In particular, does $\ker(\varphi_{*,a})=\ker (\phi_{*,a})$ for all $a\in P$?