TITLE: Grassmannian bundle theorem
QUESTION [9 upvotes]: Let's consider a vector bundle $E$ of rank $n$ over a compact manifold $X$. Consider the associated Grassmannian bundle $G$ for some $k < n$, obtained by replacing each fiber $E_x$ by $Gr(k,E_x)$.
Let's suppose that there is a full flag of subbunldes $F_1 \subset F_2 \dots \subset F_n \subset E$. I think that in this case we are able to define relative Schubert cycles on G which restrict to usual Schubert cycles on each fiber so that we can apply Leray-Hirsh theorem to deduce that $H^*(G) = H^*(X) \otimes H^*(Gr(k,n))$.

Is the reasoning above correct?
Can we still compute $H^*(G)$ in the case when the full flag of subbundles doesn't exist?

EDIT: I meant complex vector bundles and complex Grassmannians. Also the bundle can be assumed holomorphic or algebraic if it makes a difference.
EDIT: Ben in his answer mentions Serre's spectral sequence that can be used in this case. Is there a reason why it will degenerate to leave $H^*(X) \otimes H^*(Gr(k, n))$ as a result?

REPLY [3 votes]: algori's answer is completely correct. In case you want a reference, this material is in Chapter 14 of Fulton's Intersection Theory. (Except that Fulton is writing about Chow rings and you asked the question in cohomology.) In particular, the additive structure is Proposition 14.6.5 and the ring structure is 14.6.6.
I'm not sure what the best references are for the analogous statements in cohomology, but you could try tracing Fulton's references and see if they help.