Let $c\_i,d\_j <n$, be a set of integers and define
$$ M=\prod Gr(c\_i,n),\quad N=\prod Gr(d\_j,n),$$
where $Gr(k,n)$ is the grassmannian of k-planes in $\mathbb{C}^n$.
Let $E\_M=\oplus E\_i^\*$, where $E\_i$ is the pullback of tautological bundle to $M$ from the i-th component. 
Similarly, let $E\_R=\oplus E\_j^\*$, where $E\_j$ is the pullback of tautological bundle to $N$ from the j-th component.


I am intending to show that the bundle
$$E\_M\boxtimes E\_N = \oplus\_{i,j} E\_i^\* \boxtimes E\_j^\* $$
over $M\times N$ has a transverse section.
Note that each component of the dirsct sum has lots of sections. In fact, the set of
sections correspond to $n\times n$ matrices.


Even if you don't know how to answer this particular question, what are the general methods of approaching such question? How can we find whether a given bundle has transverse sections? May be some sort of numerical criterion?



How can we show that a transverse section exist
If a vector bundle on a smooth variety is globally generated then its generic section is transversal to zero. This is Bertini Theorem, so of course you need the base field to be infinite. In your special case note that the dual tautological bundles are globally generated and that pullbacks, tensor products, and direct sums of globally generated bundles are globally generated.


