Let $X$ be a smooth projective variety over $\mathbb{C}$, and $D_1, \ldots, D_n$ be a collection of simple normal crossing divisors. The divisors induce a stratification $\mathcal{T}_X$ of $X$.

Let $T_1, \ldots, T_k$ be strata and $f \colon Y\to X$ be the iterated ordered blowup of $X$ at $T_1$, then at $T_2$, ... and finally at $T_k$, and let $E_1, \ldots, E_k$ be the exceptional divisors.

The exceptional divisors induce a stratification $\mathcal{T}_Y$ of $Y$.

Questions:

Is there a way to relate the stratification $\mathcal{T}_X$ and the stratification $\mathcal{T}_Y$? For example, can the poset of $\mathcal{T}_Y$ be explicitly described from the poset of $\mathcal{T}_X$?

Let $a_1, \ldots, a_k \in \mathbb{N}$. Is there an explicit way to express the cohomological pushforward $f_*(E_1^{a_1}\cdot \cdots \cdot E_k^{a_k})$ in terms of the strata in $\mathcal{T}_X$ and the Chern classes of their normal bundles?

(For the questions to be interesting, I think one may assume that the pairwise intersection of the strata $T_1, \ldots, T_k$ is non-transversal).