First, if a semisimple simply connected group $G$ acts on a variety $X$ then any exceptional object in $D^b(coh(X))$ has a $G$-equivariant structure (proved by Elagin and by Polishchuk).
Second, indeed, as abx mentioned Kapranov has constructed a full exceptional collection on all (partial) flag varieties of type $A$ and on quadrics. Moreover, using a result of Samokhin one can construct a full exceptional collection on $X$ for $X$ having a fibration structure over $Y$ if there is a collection of objects on $X$ which restrict to a full exceptional collection on all fibers of $X$ over $Y$ and if $Y$ itself has a full exceptional collection. This allows to construct a full exceptional collection on some partial flag varieties of type $B$, $C$ and $D$ (including full flags). Using a recent result of Manivel and Faenzi one can also construct a full exceptional collection on some partial flags of type $E_6$. Finally a full exceptional collection is known on all flag varieties of type $G_2$.
And last, as Daniel Pomerleano said, there is a general construction of exceptional collections on flag varieties which produces a collection of maximal possible length on all flag varieties for groups of type $B$, $C$, and $D$ which is not known to be full. And unfortunately, it is not true that an exceptional collection generating $K_0$ is full --- there are counterexamples by Boehning-Graf vn Bothmer-Katzarkov-Sosna and by Gorchinsky-Orlov.