It's a good place to work and study, the Joshua Tree Field Station (very cool hotel).

I am still trapped by the disk with 3 stops, no punctures.

But do you understand the disk with 2 stops, how the gluing works? one disk has $k_1$ strands; another disk has $k_2$ strands, both with $2$ stops, put them together, get $k=k_1+k_2$ strands.

It is about going from $BGL(k_1)$ and $BGL(k_2)$ to $BGL(k)$. I just know $T^{k_1}$ and $T^{k_2}$ merges to $T^{k_1+k_2}$. Let's try parabolic induction: we have $$ G_1\times G_2 = GL(k_1) \times GL(k_2) \gets P=P_{k_1, k_2} \to GL(k_1+k_2)=G $$

Assume everything is fine, then we have $$ BP \to B(G_1 \times G_2), \quad BP \to BG $$ The first one is saying, if you have a principle $P$ bundle $E$ over something $M$, then you can build the associated bundle of $E \otimes_P L$.

I guess the corresponding construction on sheaf is just doing pull-push along the parabolic induction.

Is there classifying space for two step flags? well, one needs to say the ranks, so it would be the $BP_{k_1, k_2}$.