If my vassals have cultural buildings for their culture, do I get the benefits when I levy?

Okay, so I understand that cultural buildings give bonuses to troops. e.g. Irish Gallowglass IV means +60% to my HI Defense. (I'm less clear on whether that only applies to troops from that specific levy, or to all of my troops everywhere.)

Suppose I have heated discussion with the neighboring Scots and I end up governing a county with Schiltron Barracks III (+45% def to pikemen). Obviously, if I change the county's culture to Irish, the Schiltron barracks will get snuffed. Do my pikemen gain the advantages of that Schiltron building? Do the pikemen levvied from that county (either my own, or liege levy) get the benefit?

I'm trying to figure out if there's any reason at all to preserve the culture of where I conquer. Being able to put Schiltron and Gallowglass into a single column seems like a great way to tell England where they can shove it, but I see conflicting accounts of what happens when I (an Irish intellectual) am liege to someone else (a Scottish gentleman)...