Let $(f_U,f_A,f_V):(U\leftarrow A\rightarrow V)\to (U'\leftarrow A'\rightarrow V')$ be a morphism of diagrams of topological spaces and let $f:U\sqcup_A V\to U'\sqcup_{A'} V'$ be the resulting morphisms for the pushouts.
Suppose I know the (weak) homotopy type of the homotopy fibres of $f_U$, $f_A$ and $f_V$. Is there any chance to say something about the homotopy fibre of $f$?
To be more precise: In my special situation, I know that $f_U$, $f_A$ and $f_V$ are all coverings and I would be happy if I could conclude that $\pi_2(\mathrm{hofib}(f))=0$.