Suppose that we have a map between two pushout diagrams of topological spaces 
$$ [A\_{1}\leftarrow A\_{0}\rightarrow A\_{2} ] \rightarrow [B\_{1}\leftarrow B\_{0}\rightarrow B\_{2} ]$$
such that for any $i\in\{0,1,2\}$, $A\_{i}\rightarrow B\_{i}$ is a trivial cofibration of topological spaces. And suppose that all spaces are CW-complexes. Moreover all maps $A\_{i}\rightarrow A\_{j}$ and $B\_{i}\rightarrow B\_{j}$ are supposed to be surjective. What can we say about the induced map $$ colim [A\_{1}\leftarrow A\_{0}\rightarrow A\_{2} ] \rightarrow colim[B\_{1}\leftarrow B\_{0}\rightarrow B\_{2} ]$$ e.g. is it a weak equivalence ?



Pushout of spaces
Take $B\_0$ to be an annulus. Let $B\_1$ be the space obtained by collapsing the left half of the inner circle to a point, and let $B\_2$ be obtained by collapsing the right half of the inner circle to a point. Let $A\_0$, $A\_1$ and $A\_2$ all be the outer circle of $B\_0$. Then the $A$-pushout is a circle whereas the $B$-pushout is a disc.


You could ask more questions along these lines, and it might take some ingenuity to find counterexamples, but they will always be there. You will almost never have good homotopical control over pushouts unless at least one arm of the pushout is a cofibration, or you have very specific information about the spaces involved.


