I agree with Allen that (a) the question is ambiguously phrased, and (b) the most interesting interpretation, likely, is not the one intended by the OP. Having said that, my own interpretation is that the OP wants to know how to compute the pullback of the Schubert classes for a morphism between Grassmannians. The short answer is that Giambelli allows to express all Schubert classes as polynomials in the special Schubert classes. So the OP only needs to compute the pullbacks of the special Schubert classes.

Of course that raises the question, what are the possible values for the pullbacks of the special Schubert classes, or, equivalently, what are the K-theory classes of globally generated, locally free sheaves of rank $r+1$. By analogy with Hartshorne's conjecture, I expect the K-theory class is either a sum of $r+1$ (globally generated) invertible sheaves or else the sum of one (globally generated) invertible sheaf and a (globally generated) twist of the universal rank $r$ locally free sheaf by an invertible sheaf.