First suppose we have three smooth manifolds $M_1$, $M_2$ and $N$ with
smooth transversal maps $p_1: M_1 \rightarrow N$ and $p_2: M_2 \rightarrow N$
then its a well known fact that the categoric universal pullback 
(in the category of smooth manifolds)
$$M_1 \times_{p_1 N p_2} M_2$$ is a smooth manifold, too. Hence we can say
that transversal pullbacks exist in this category.

Next we have a little more structure on $M_1$ and $M_2$ in that we assume
each $p_1 : M_1 \rightarrow N$ and $p_2 : M_2 \rightarrow N$ to be a locally
trivial smooth fiber bundle. In this case the categoric universal pullback
$M_1 \times_{p_1 N p_2} M_2$ is again a fiber bundle over $N$ called the fiber product
$M_1 \oplus M_2$ or $M_1 \times_N M_2$. 

(But observe here that since $N$ is
just a manifold and not a fiber bundle this is not the universal pullback in the 
category of smooth fiber bundles !)

This again is well known.

But now we assume in addition that there is a fiber bundle structure $p_N: N \rightarrow L$. 
Then $M_1$, $M_2$ and $N$ are fibered manifolds
and my first question is, if the categoric pullback 
$M_1 \times_{p_1 N p_2} M_2$ is defined in the category of local trivial fiber bundles?

In that case it should be a fiber bundle over $L$.

Suppose we restrict the pullback to vector bundles, then is it again a vector bundle
or is it a second order vector bundle or is the linear structure lost?