You mentioned the degeneracy subset definition. First lets re-establish the Grassman case as an example. Let $BU(n)=Gr_n(\mathbb{R}^{\infty})$ and take $e_i=(0,\dots 1, \dots )$ in the $i$-th position. Then define a vectorfield of the canonical bundle by $pr_{V}(e_1)$ at point $V$. Then the vanishing set is $Gr_n(\langle e_1^{\perp}\rangle)$. The fact that this is "generic" is easily checked, so that $c_1=[Gr_n(\langle e_1^{\perp}\rangle)]$. Now, we can take more vectorfields, $v_i$, defined analogously, and the dependant sets. But since $v_i$ are always orthogonal, this is simply gonna be the cap product, by transverality of $[Gr_n(\langle e_1^{\perp}\rangle)]$, or simply $c_i=[Gr_n(\langle (e_1,\dots e_i)^{\perp},\rangle)]$. This is easier to see in the more familiar setting of $BU(1)$. We have that $H^1(BU(1))=H^1(\mathbb{C}P^{\infty})=Hom(H_1(\mathbb{C}P^{\infty}), \mathbb{Z})=[\mathbb{C}P^{\infty-1}]^*$ and likewise, $H^i(BU(1))=[\mathbb{C}P^{\infty-i}]^*$, where these terms denote the limit of the inclusiosn $\mathbb{C}P^{n-i}\to \mathbb{C}P^n$. The situation for Stiefel-Whitney classes is similar.

Now for the general flag manifold, say $F=F(d_1,\dots d_i, n)$. Then take $pr_{V_i}(e_1)$, which gives a deneracy set of $[F(d_1, \dots d_{i}, n-1)]$ and so on, and we can proceed exactly as in the Grassmanian case getting $[F(d_1, \dots d_{i}, n-l)]=c_l$. This noteably agrees on the dot with the canonical maps involving other flag manifolds. This gives exactly the result we want, since the map $F\to BU(d_i)$ induces the top tautological bundle, and thus the Chern class is just the pullback of the original above class. Similar results hold for all other tautological bundles on the flag.

You mentioned the degeneracy subset definition. First lets re-establish the Grassman case as an example. Let $BU(n)=Gr_n(\mathbb{R}^{\infty})$ and take $e_i=(0,\dots 1, \dots )$ in the $i$-th position. Then define vectorfields of the canonical bundle by $v_i=pr_{V}(e_i)$ at point $V$. Then the degenerancy set is $X=\{V| det(pr_V(e_i))\neq 0\}$. The fact that this is "generic" is easily checked, so that $c_1=[X]$. Now since $v_i$ are always orthogonal, this is simply gonna be the cap product, by transverality of $\{V| det(pr_V(e_i))\neq 0\}$, or simply $c_i=[\{V|rank(pr_V(e_k))<i, k\le i\}]$, or the set such that the first $i$ $v_i$ are linearly dependant. This is easier to see in the more familiar setting of $BU(1)$. We have that $H^1(BU(1))=H^1(\mathbb{C}P^{\infty})=Hom(H_1(\mathbb{C}P^{\infty}), \mathbb{Z})=[\mathbb{C}P^{\infty-1}]^*$ and likewise, $H^i(BU(1))=[\mathbb{C}P^{\infty-i}]^*$, where these terms denote the limit of the inclusiosn $\mathbb{C}P^{n-i}\to \mathbb{C}P^n$. The situation for Stiefel-Whitney classes is similar.

Now for the general flag manifold, say $F=F(d_1,\dots d_i, n)$. Then take $v_j=pr_{V_i}(e_j)$, which gives a deneracy set of the dependant sets of the $v_j$, and so on, and we can proceed exactly as in the Grassmanian case getting $c_l=\{V|rank(v_j)<l\}$. This noteably agrees on the dot with the canonical maps involving other flag manifolds. This gives exactly the result we want, since the map $F\to BU(d_i)$ induces the top tautological bundle, and thus the Chern class is just the pullback of the original above class. Similar results hold for all other tautological bundles on the flag.