Let $ f \colon E \to B $ be a morphism between (weak) homotopy types such that each (homotopy) fiber has a homotopy type of some finite CW complex.
Then does $ \Sigma f \colon \Sigma E \to \Sigma B $ have the same property?



Is the suspension of a finite fibration again finite?
Assuming you work with unpointed spaces (but the example can easily be adapted to the pointed case) the map $1 \to 2$ gives a counterexample : its fiber are $1$ and $\varnothing$ so they are both finite, but on applying the suspension you get $1 \to S^1$ whose homotopy fibers identitifes with $\mathbb{Z}$.


