Bundle map inducing isomorphism on fibers - when is it an isomorphism?

Question

Suppose $\begin{smallmatrix}X\\ \downarrow\\ U \end{smallmatrix}\to\begin{smallmatrix}Y\\ \downarrow\\ U \end{smallmatrix}$ is a bundle map inducing homeomorphisms on fibers.

1. What are some general conditions for this map to be a bundle isomorphism?
2. What is an example of such a map which is not a bundle isomorphism?
3. Is this map always an isomorphism when restricting to the category of locally connected spaces?

Edit. The bundles are not assumed locally trivial. Assume also the base $U$ is connected.

Answer 1

If $X,Y$ are trivial bundle, say $X = Y = U \times F $ then your map is on the form $(u,\lambda) \mapsto (u,\lambda,f_u(\lambda))$ where $f : U \to \text{Homeo}(F)$ is a continuous map. It is clear that this map is invertible with inverse $(u,f_u^{-1}(\lambda))$.

For the general case, we want to see if $f$ is an homeomorphism i.e if $f^{-1}$ is continuous, but continuity is a local condition so by the previous paragraph you are done.