Asthma - eosinophil movement into airways

[Greippi]

I'm confused about the late phase of the asthma response in which eosinophils are involved.

 

My understanding is that eosinophils infiltrate the airways from the surrounding capillaries, causing inflammation. In time, epithelial cells are damaged and become hardened due to repetitive remodelling.

To do this, they must first cross the endothelial cells that make up the capillaries via the adhesion cascade using various selectins (such as P-selectin on an endothelial cell), and integrins (such as VLA-4).

 

But then how do they cross the epithelial cells to get inside the airways? Is there a similar adhesion cascade?