I'm not aware of any generalization that is strong enough to compute the cohomology of such bundles completely, but you can at least use the Borel--Weil--Bott theorem to get some vanishing results. This was already done by Bott in his 1957 Annals paper to show that the cohomology of the tangent bundle $T$ of $G/P$ vanishes in degree > 0. The first step of this computation is to note that the action of $P$ on the tangent space at $P/P$ can be identified, as a $P$-module, with $\mathfrak g/ \mathfrak p$ (with $P$ acting via Ad). Consequently, the short exact sequence of $P$-modules
$$ 0 \to \mathfrak p \to \mathfrak g \to \mathfrak g / \mathfrak p \to 0 $$
yields a short exact sequence of $G$-equivariant vector bundles
$$ 0 \to G \times_P \ \mathfrak p \to G \times_P \ \mathfrak g \to T \to 0 $$
on $G/P$. The middle bundle is trivial, and so we get an isomorphism $H^q(G/P, T) \cong H^q(G/P, G \times_P \ \mathfrak p)$ for all $q>0$.

It remains to compute $H^q(G/P, G \times_P \ \mathfrak p)$. For this, a vanishing result of the Borel--Weil--Bott type is useful. This result states that if you have a $G$-equivariant vector bundle $\mathcal V = G \times_P V$ on $G/P$ then $H^q(G/P, \mathcal V) = 0$ unless $q$ is equal to the index of some nonsingular $\mu + \rho$, where $\mu$ is one of the highest weights of $V$.

For the details, you can either try Bott's original paper, or checkout section 11 of [this paper][1] of Dennis Snow. Another paper of potential interest is
> P.A. Griffiths, *Some geometric and analytic properties of homogeneous complex manifolds. I. Sheaves and cohomology*, Acta Math. **110** (1963), 115–155.


  [1]: http://www.nd.edu/~snow/Papers/HomogVB.pdf